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 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
Therefore, each outcome in the sample space will consist of two numbers — one for each die.
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 possible outcomes, the total number of possible outcomes for this experiment is
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: Here, and represent the white marbles, and and represent the black marbles.
Since both marbles are randomly drawn, each outcome of the event will consist of two labels. For the first marble, there are possible outcomes. For the second marble, there are possible outcomes because one marble has already been removed from the bag.
Since the Diego draws a marble first and Emily draws a marble after him, the order in which the marbles are drawn matters. Therefore, the outcomes and are different. The following table lists all the possible outcomes for the event of drawing two marbles from the bag. Consequently, there are a total of possible outcomes in the sample space.
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.
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 of the Complements:
|picking a prime number||picking a non-prime number|
|picking an odd number||picking an even number|
|picking a multiple of||picking a number that is not a multiple of|
In a similar way, the outcomes of the remaining events can be written. Finally, the outcomes of are the outcomes in that are not in Similarly for the outcomes of and
- event of picking a prime number
- event of picking an odd number
Therefore, is the event of picking a prime and odd number.
|Number||Is it prime?||Is it odd?|
- event of picking an odd number
- event of picking a number that is not a multiple of
Therefore, is the event of picking either an odd number or a number that is not multiple of
|Number||Is it odd?||Is not a multiple of|
Consequently, Notice that is the only element of that is not in since it satisfies neither nor
Next, draw three sets representing the events and 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
Since the experiment consists of rolling two dice and each die has possible outcomes, there is a total of possible combinations. Calculate the sum of the outputs for each combination.
The in the second row and third column represents the event of rolling a on the first die and a 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
Therefore, to find the experimental probability obtained by Ramsha, divide the number of successes she got by the number of trials she conducted. These two numbers can be deduced from the given diagram.
The probability of drawing a club from a standard deck of cards is 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.
This formula is useful when calculating the probability of the complement of an event is easier than calculating the probability of the event itself. Then, the probability of the event is calculated as follows.
Let be an event, be its complement, and be the sample space. By the definition of the complement, the union of an event and its complement is equal to the entire sample space. Because and represent the same event, their probabilities are equal. Since the complement of consists of the outcomes that are not in events and are disjoint. By the Addition Rule of Probability, the probability of the union is the sum of the individual probabilities of each of the events. Now the Transitive Property of Equality can be applied to the equalities. Additionally, the probability of the entire sample space is equal to By applying the Transitive Property of Equality once more, the rule is proven.
Using the Subtraction Property of Equality, the formula for the probability of is obtained.
Applying this formula, the probability of drawing a spade, heart, or diamond can be computed.
Dylan cut out squares of paper and wrote a number from to on each square. He then put the papers in a bag and asked his dad to choose a paper at random.
Start by remembering what the probability of an event is. Let be the given event, picking a number that is not a multiple of However, counting the numbers from to that are not multiples of can be tedious. In this case, it is worth considering the complement of
|Picking a number that is not a multiple of||Picking a number that is a multiple of|
Since there are three doors, there are total possible outcomes, of which only is favorable. Knowing this, the chance that Tearrik will choose the wining door can be calculated.
In Part A, the probability of picking the winning door was found to be about Substituting this value into the previous equation, the probability of picking a door with a sheep will be obtained.