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| 13 Theory slides |
| 14 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Below, some basic definitions of probability are examined.
An outcome is a possible result of a probability experiment. For example, when rolling a six-sided die, getting a 3 is one possible outcome.
An event is a combination of one or more specific outcomes. For example, when playing cards, an event might be drawing a spade or a heart. For this event, one possible outcome is drawing the A♠ or drawing the 7♡.
However, these are not the only outcomes of this event. All the possible outcomes that satisfy the event are listed below.
The sample space of an experiment is the set of all possible outcomes. For example, when flipping a coin, there are two possible outcomes: heads, H, or tails, T. Therefore, the sample space is {H,T}.
Here, the sample space is shown in a tree diagram. Each row represents the possible outcomes of a toss. When the coin is flipped another time, the tree diagram gets another row with the possible outcomes.
In this case, the sample space has 4 possible outcomes.
For each of the following experiments, list the possible outcomes in the sample space and count the total number of outcomes.
Outcomes:
Consequently, the sample space is the set {1,2,3,4,5,6}.
To list all the possible outcomes, create all possible combinations by determining one outcome for the first die and then varying the outcome of the second die. Then, change the outcome of the first die and repeat the process.
Since each die has 6 possible outcomes, the total number of possible outcomes for this experiment is 6⋅6=36.
Paulina bought two white and three black marbles, all of different sizes, and put them in a bag. When she got home, her little brother Diego and sister Emily asked her to give them two marbles. Paulina agreed but told them to draw one marble each without looking inside the bag. Diego drew the first marble, then Emily.
Outcomes: {W1,W2} and {W2,W1}
Since both marbles are randomly drawn, each outcome of the event will consist of two labels. For the first marble, there are 5 possible outcomes. For the second marble, there are 4 possible outcomes because one marble has already been removed from the bag.
Sometimes more than one event can be involved in an experiment. In such cases, it is important to know how to determine the union or intersection of the events.
A or Bor
A∪B.
A and Bor
A∩B.
Also, there may be situations where it is easier to determine which outcomes do not satisfy an event rather than determining which outcomes do. For such cases, the following concept will be useful.
Outcomes: A∩B={3,5,7}
Outcomes: B∪C′={1,2,3,4,5,7,8,9}
Event | Complement |
---|---|
A: picking a prime number | A′: picking a non-prime number |
B: picking an odd number | B′: picking an even number |
C: picking a multiple of 3 | C′: picking a number that is not a multiple of 3 |
A - event of picking a prime number
B - event of picking an odd number
Therefore, A∩B is the event of picking a prime and odd number.
Number | Is it prime? | Is it odd? |
---|---|---|
1 | No | Yes |
2 | Yes | No |
3 | Yes | Yes |
4 | No | No |
5 | Yes | Yes |
6 | No | No |
7 | Yes | Yes |
8 | No | No |
9 | No | Yes |
Consequently, A∩B={3,5,7}.
B - event of picking an odd number
C′ - event of picking a number that is not a multiple of 3
Therefore, B∪C′ is the event of picking either an odd number or a number that is not multiple of 3.
Number | Is it odd? | Is not a multiple of 3 |
---|---|---|
1 | Yes | Yes |
2 | No | Yes |
3 | Yes | No |
4 | No | Yes |
5 | Yes | Yes |
6 | No | No |
7 | Yes | Yes |
8 | No | Yes |
9 | Yes | No |
Consequently, B∪C′={1,2,3,4,5,7,8,9}. Notice that 6 is the only element of U that is not in B∪C′ since it satisfies neither B nor C′.
Next, draw three sets representing the events A, B, and C and write down the outcomes of each event inside the corresponding set.
Comparing the outcomes of the three events, some conclusions can be drawn.
With this information, the Venn diagram can be drawn.
Since an event is a combination of possible outcomes of an experiment, in some cases the event happens rarely, while in others it happens frequently. This frequency depends on the experiment and the event itself.
The probability of an event occurring can be determined both theoretically and experimentally. Theoretical probability expresses the expected probability when all outcomes in a sample space are equally likely. In comparison, experimental probability is expressed by analyzing data collected from repeated trials of an experiment.
When all outcomes in a sample space are equally likely to occur, the theoretical probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes.
P(event)=Number of possible outcomesNumber of favorable outcomes
When an experiment is performed, the results may be a little different from what was expected. In other words, slightly different results may be obtained from what the theoretical probability predicted.
Experimental probability is the probability of an event occurring based on data collected from repeated trials of a probability experiment. For each trial, the outcome is noted. When all trials are performed, the experimental probability of an event is calculated by dividing the number of times the event occurs, or its frequency, by the number of trials.
Ramsha and Mark conducted an experiment consisting of rolling two dice and adding their results. The following diagram shows the numbers obtained in each roll.
Consider the event of getting a result greater than or equal to 8.
The 5 in the second row and third column represents the event of rolling a 2 on the first die and a 3 on the second die. The other outcomes can be calculated in the same fashion. Next, highlight the outcomes satisfying the given event, that the sum of the dice is greater than or equal to 8.
Favorable outcomes=15, Total outcomes=36
ba=b/3a/3
Calculate quotient
Round to 2 decimal place(s)
Number of Successes=2, Number of Trials=7
Calculate quotient
Round to 2 decimal place(s)
Number of Successes=6, Number of Trials=9
ba=b/3a/3
Calculate quotient
Round to 2 decimal place(s)
The probability of drawing a club from a standard deck of cards is 0.25. Knowing this, what is the probability of drawing a spade, heart, or diamond if a card is drawn randomly?
To figure it out, instead of counting the favorable outcomes, the complement rule can be used.
The sum of the probability of an event and the probability of its complement is equal to 1.
P(A)+P(A′)=1
P(A)+P(A′)=1
Applying this formula, the probability of drawing a spade, heart, or diamond can be computed.
Dylan cut out 100 squares of paper and wrote a number from 1 to 100 on each square. He then put the papers in a bag and asked his dad to choose a paper at random.
What is easier to count, the numbers that are not multiples of 5 or the numbers that are multiples of 5? Note that these sets of numbers are the complements of each other.
Event | Complement |
---|---|
Picking a number that is not a multiple of 5. | Picking a number that is a multiple of 5. |
Favorable outcomes=50, Total outcomes=70
ba=b/10a/10
Calculate quotient
Round to 2 decimal place(s)
Jordan is playing craps which requires a pair of dice. When playing craps, both dice are thrown and their outcomes are added.
Let's draw all the possible combinations when rolling two dice.
There are multiple combinations that add up to the same number. Remember, we cannot count an outcome multiple times. Therefore, the sample space is as follows. {2,3,4,5,6,7,8,9,10,11,12} The sample space has 11 outcomes.
To determine the probability of rolling an even number, we will highlight all even combinations in the table made earlier. Remember, an even number is divisible by 2.
We see that 18 combinations are even. Since the sample space is 6 by 6, the total number of possible combinations is 6* 6= 36. Now we can calculate the probability by dividing the number of favorable combinations by the number of possible combinations. P=18/36= 1/2 The probability of rolling an even number is 12.
Let's list all of the dice combinations that give a difference of 3. & Die 1-Die 2 [-1.1em] &D1 - D2 = 4-1 = 3 &D1 - D2 = 5-2 = 3 &D1 - D2 = 6-3 = 3 [1em] & Die 2-Die 1 [-1.1em] &D2 -D1 = 4-1 = 3 &D2 -D1 = 5-2 = 3 &D2 -D1 = 6-3 = 3 Let's highlight all of these combinations in the table we made earlier.
Of the 36 possible combinations, 6 of them give a favorable outcome. Now we can calculate the probability that the difference between the dice is 3. P=6/36= 1/6
The most likely outcome is the number in the sample space that occurs most frequently. If we again look at the table with all the possible combinations for the dice, we see that 7 is the most frequent outcome.
There are 6 different combinations that give a seven. With this information, we can calculate the probability of rolling this outcome. P=6/36= 1/6
A manufacturer of computers tests 2000 computers and finds that 8 of them have defects.
From the exercise, we know that of the 2000 computers tested, 8 have defects. If we divide the number of defective computers by the total number of tested computers, we get the probability of a computer having a defect.
Consequently, there is a 0.4 % chance that a computer picked at random has a defect.
To predict the number of computers with defects in a shipment of 100 000, we must multiply the probability found earlier by the number of computers in the shipment. 100 000(0.004)=400 About 400 computers in the shipment could be defective.
Find the total number of possible outcomes for each of the following situations.
A card is drawn from a standard deck of cards. |
We flip a coin and draw a marble from a bag containing two purple marbles and one gold marble. |
We draw three marbles, one at a time and without replacing them, from a bag containing three green marbles and four blue marbles. |
A standard deck has 52 playing cards. Therefore, the number of possible outcomes when drawing a card from one is 52.
In this case, we have a coin and a bag of marbles. A coin has two outcomes, heads and tails. This produces two outcomes.
Next up, we have to choose a marble. This can be purple or gold. Notice that we are not concerned with probabilities to figure out the possible outcomes. Only the number of possible outcomes is important, which means we will disregard the fact that there are more purple marbles than gold marbles in the bag.
Therefore, the number of possible outcomes in the sample space is 4. {H, P}, {H, G }, {T, P}, {T, G}
As in the previous situation, we are not concerned with probabilities at this point. Rather, we want to list all the possible outcomes. We can make another tree diagram to show all the possibilities.
As we can see, we have a total of 8 possible outcomes. {G, B, B}, {G, B, G}, {G, G, B}, {G, G, G}, {B, B, B}, {B, B, G}, {B, G, B}, {B, G, G}
Jordan rolls a regular six-sided die.
Calculate the probability of the following events happening. Write the answers as a fraction.
To calculate the probability of an event, we have to divide the number of favorable outcomes by the number of possible outcomes. P=Number of favorable outcomes/Number of possible outcomes In this case, Jordan has a regular six-sided die and we want to find the probability of her rolling at least a 2. This means we cannot roll a 1.
We have 5 favorable outcomes of 6 possible outcomes. Now we can determine the probability. P(At least a2)=5/6
Let's find all the possible even outcomes on a die.
We have 3 favorable outcomes out of 6 possible outcomes. Now we can determine the probability. P(Even)=3/6=1/2
The urn below contains three red, four blue, and five yellow balls.
Kriz pulls out a ball without looking. Give the probability as a fraction that the ball is...
Probability is calculated by dividing the number of favorable outcomes by the number of possible outcomes. P=Number of favorable outcomes/Number of possible outcomes We have 4 blue balls, 3 red balls, and 5 yellow balls. blue balls:& && 4 red balls:& && 3 yellow balls:& && 5 [-1.2em] total:& &&12 We want to know the probability of picking a blue ball. With the given information, we can determine the probability of picking a blue ball. P( Blue) = 4/12=1/3
This time, we must divide the sum of the blue and yellow balls by the total number of balls. From the previous section we know that 4 balls are blue and 5 are yellow, for a total of 9 balls. Now we can determine the probability. P(BlueorYellow) = 9/12=3/4 The probability of picking a blue or a yellow ball is 34.
The event of picking a ball that is neither blue nor yellow is the complement of picking a blue or yellow ball. We already calculated the probability of picking a blue or yellow ball in the previous section. P(Blue or Yellow) = 3/4 Therefore, the probability of picking a ball that is neither blue nor yellow can be found using the complement rule.
The probability of picking a ball that is neither blue nor yellow is 14.
Let's think about the statements one at the time.
A person does not have to have gone to college to have a high income. However, it still increases the possibility of securing a high-paying job if they have. The most appropriate probability would be 0.7. This is completely arbitrary, by the way — someone can definitely succeed without having gone to college!
Ants are not able to fly a jet created for a human. The probability of this is therefore 0.
The ace of spades is one card out of 52 cards in a standard deck. To calculate the probability of being dealt this card, we divide 1 by 52. P=1/52≈ 0.02 The probability is about 0.02.
There are two possible outcomes when we flip a coin, either heads or tails. Therefore, the probability of either of them happening must be 1.
If the person does not win $10, they will have drawn either a no prize ticket or a $20 prize ticket. They therefore have the following complement. $20 prize and No prize