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 or drawing the
However, these are not the only outcomes of this event. All the possible outcomes that satisfy the event are listed below.
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.
Consequently, the sample space is the set
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.
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 and Bor
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.
|A: picking a prime number||picking a non-prime number|
|B: picking an odd number||picking an even number|
|C: picking a multiple of 3||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, is the event of picking a prime and odd number.
|Number||Is it prime?||Is it odd?|
B - event of picking an odd number
- event of picking a number that is not a multiple of 3
Therefore, 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|
Consequently, Notice that 6 is the only element of U that is not in since it satisfies neither B nor
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 shows the expected probability when all outcomes in a sample space are equally likely, whereas experimental probability is based on data collected from repeated trials of an experiment.
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 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.
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.
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.
|Picking a number that is not a multiple of 5.||Picking a number that is a multiple of 5.|