12.4 Probability

Learning Objectives

After Chapter 12.4, you will be able to:

• Define independence, mutual exclusivity, and exhaustiveness
• Calculate the probability of an event, or of co-occurrence of multiple independent events
  

Probability is usually tested on the MCAT in the context of a science question, rather than being tested on its own. In particular, genetics questions involving the Hardy-Weinberg equilibrium and Punnett squares are common applications of probability. Probability also underlies statistical testing, which we will investigate in the next section.

Independence, Mutual Exclusivity, and Exhaustiveness

In probability problems, we must first determine the relationship between events and outcomes. For events, we are most interested in independence or dependence. Conceptually, independent events have no effect on one another. If you roll a die and get a 3, then pick it up and roll it again, the probability of getting a 3 on the second roll is no different than it was before the first roll. Independent events can occur in any order without impacting one another.

Key Concept

Independent events do not impact each other, so their probabilities are never expected to change.

Dependent events do have an impact on one another, such that the order changes the probability. Consider a container with five red balls and five blue balls. The probability that one will choose a red ball is If a red ball is indeed chosen, then the probability of drawing another red ball is If, however, a blue ball is chosen, then the probability of drawing a red ball is In this way, the probability of the second event (getting a red ball on the second draw) is indeed dependent on the result of the first event.

We are also concerned with whether events are mutually exclusive or not. This term applies to outcomes, rather than events. Mutually exclusive outcomes cannot occur at the same time. One cannot flip both heads and tails in one throw, or be both ten and twenty years old. The probability of two mutually exclusive outcomes occurring together is 0%.

Finally, we must consider if a set of outcomes is exhaustive or not. A group of outcomes is said to be exhaustive if there are no other possible outcomes. For example, flipping heads or tails are said to be exhaustive outcomes of a coin flip; these are the only two possibilities.

Calculations

For independent events, the probability of two or more events occurring at the same time is the product of their probabilities alone

P(A ∩ B) = P(A and B) = P(A) × P(B)

Equation 12.6

For example, the probability of getting heads on a coin flip twice in a row is the same as the probability of getting heads the first time times the probability of getting heads the second time, or 0.5 × 0.5 = 0.25. The probability of two independent events co-occurring is shown diagrammatically in Figure 12.4.

Figure 12.4. Probability of Two Independent Events Co-Occurring P(A and B) = P(A) × P(B)

The probability of at least one of two events occurring is equal to the sum of their initial probabilities, minus the probability that they will both occur.

P(A ∪ B) = P(A or B) = P(A) + P(B) − P(A and B)

Equation 12.7

Key Concept

In probability, when using the word:

• and—multiply the probabilities
• or—add the probabilities (and subtract the probability of both happening together)

Example:

In a certain population, 10% of the population has diabetes and 30% is obese. If 7% of the population has both diabetes and obesity, are these events independent? If one chose an individual at random from this population, what would be the probability of that patient having at least one of the two conditions?

Solution:

With the numbers given, these events cannot be independent. For independent events, P(A and B) = P(A) × P(B) = P(having diabetes) × P(being obese) = 0.1 × 0.3 = 0.03. In this population, the probability of having diabetes and being obese is 0.07.

To determine the probability of the individual having at least one of the conditions, we use the “or” equation:

P(A or B) = P(A) + P(B) − P(A and B) = 0.1 + 0.3 − 0.07 = 0.33 or 33%

MCAT Concept Check 12.4

Before you move on, assess your understanding of the material with these questions.

1. Assume the likelihood of having a male child is equal to the likelihood of having a female child. In a series of ten live births, the probability of having at least one boy is equal to:

2. Define the following terms:

   • Independence:
     
   • Mutual exclusivity:
     
   • Exhaustiveness: