Module 03 Determining Probability

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Module 03 Determining Probability  Determining Probability Compound Events In statistics, it is common to look at combining more than one event to find the probability. This section will look at finding the probability that either event A or event B occurs and the probability that both event A and event B occur. For the “OR” probability, addition and subtraction will be used to determine probability. Whereas, in the “AND” probability, multiplication will be used to determine probability.

Addition Rule We use the addition rule when trying to find the probability of an event A or an event B occurring. The word “or” in the addition rule is an “inclusive or” meaning that you can either have event A occur, event B occur, or both events occur. To find this probability, we take the number of ways event A can occur and add it to the number of ways event B can occur, but only so that each outcome is counted once. The formula is given below. The reason the probability of A and B is subtracted at the end is to take away any double-counting due to overlapping outcomes between events A and B (Triola & Iossi, 2018). Examples of Addition Rule Using the procedure of rolling a six-sided die once, what is the probability of the following compound events? Example 1 - What is the probability that you will roll an even number or number greater than 4? Recall that rolling a six-sided die produces six possible outcomes of the procedure: {1, 2, 3, 4, 5, 6}. The two events being looked at in this experiment are rolling an even number which has three outcomes: {2, 4, 6}, and rolling a number greater than four, which contains two outcomes: {5, 6}. Notice that there is one outcome, which is in both events: {6}. This means that the compound event of A and B would have 1 outcome. Using the formula above, the probability of rolling an even number or number greater than 4 would be

Example 2 - What is the probability that you will roll a five or a number less than 3? Recall that rolling a six-sided die produces six possible outcomes of the procedure: {1, 2, 3, 4, 5, 6}. The two events being looked at in this experiment are rolling a five, which only has one outcome: {5} and rolling a number less than three, which contains two outcomes: {1, 2}. Notice that they do NOT have any outcomes in common, so A and B would be empty. Using the formula above, the probability of rolling a five or a number less than 3 would be Notice in the example above that the two events of rolling a five and a number less than 3 resulted in the probability of 0. This indicates that the two events could never occur at the same time. When two events, A and B, cannot occur at the same time, these events are disjoint (or mutually exclusive ). When two events are disjoint, the addition rule simply becomes since the probability of A and B would be zero (Triola & Iossi, 2018).

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Multiplication Rule The second way to combine two events is with the term “and.” This indicates that both events must occur. This can be one event occurring right after the other, or both events occurring at the same time. To determine the formula to use for “and” probability depends if the two events are independent or dependent. Independent Events If two events are independent, the outcome of one event does not affect the probability of the second event. For example, rolling a die twice in a row would be independent events since the outcome of the first die roll will not affect the outcome of rolling the die a second time. For independent events, the probability of event A and event B occurring would be the probability of each event multiplied together (Triola & Iossi, 2018). Example - Independent Events Find the probability of rolling five two times in a row on a six-sided die. Since the event of rolling a die does not affect the outcome of rolling a second die, these two events would be considered independent. The probability of rolling a five on a six-sided die two times in a row would be the following:

Dependent Events If two events are not independent, they would be considered dependent. When the outcome of one event affects the probability of a second event, these events would be considered dependent. For example, the probability of pulling two cards from a deck of cards would be dependent because the outcome of the first card will affect the outcome of the second card. For dependent events, the probability of the second event, B, is dependent on the first event, A. The notation for “B given A” would be written as P(B|A) . To find the probability of two dependent events which are dependent, we use the following formula (Triola & Iossi, 2018). Example - Dependent Events What is the probability of drawing two kings in a row from a standard deck of cards? Since the cards are being pulled from the deck without replacement, these two events would be considered dependent. The probability of the first event (pulling a king) would be 4/52 since there are four kings in the deck. However, the probability of the second event (pulling a second king) is contingent on the fact that you pulled a king the first round. This decreases the number of kings in the deck to 3 and the number of cards overall to 51. Thus, P(2 nd king | 1 st king) = 3/51. The overall probability of both of these events occurring would be:

When asked to find an “and” probability, start by defining if the events are dependent or independent. From there, use the appropriate formula. Another hint to decipher the difference is that independent events can use the phrase “with replacement,” but dependent events can use the phrase “without replacement.” In our examples, notice that these two statements were implied, but not explicitly stated. Be careful in probability to read the problem carefully, but also perform some examples of outcomes to differentiate between these types of events. Additional Reading - Poker Luck or Skill? Poker is a game that requires knowledge of probability and a bit of luck. Read this article, Is poker a game of luck or skill? to learn the difference between skill and luck while playing Poker. References Schoenberg, F. P. (2018). Is poker a game of luck or skill? Significance, 15 (6), 30- 33. https://doi.org/10.1111.j.1740-9713.2018.01212.x Triola, M. F., & Iossi, L. (2018). Elementary statistics (13 th ed.). Pearson.

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