Probability Using Permutations and Combinations Math in Our World Section 11.5 Probability Using Permutations and Combinations

Learning Objectives Compute probabilities using combinations. Compute probabilities using permutations.

Events with Large Possibilities When the number of possibilities gets larger, the combination and permutation rules from will be our best friends. Our general game plan will be to use these rules to find the number of outcomes that satisfy a certain event, as well as the total number of outcomes in the sample space. Then we can divide the first number by the second to obtain the probability of the event occurring.

EXAMPLE 1 Using Combinations to Compute Probability Stacy has the option of selecting three books to read for a humanities course. The suggested book list consists of 10 biographies and five current events books. She decides to select the three books at random. Find the probability that all three books selected will be current events books. SOLUTION Since there are five current events books and Stacy will need to select three of them, then there are 5C3 ways of doing this.

EXAMPLE 1 Using Combinations to Compute Probability SOLUTION There are 10 ways to choose 3 current events books. The total number of outcomes in the sample space is 15C3 since she has to select three books from 15 books. The probability of selecting three current events books is

EXAMPLE 2 Using Combinations to Compute Probability What is the probability of getting 4 aces when drawing 5 cards from a standard deck of 52 cards? SOLUTION First, we’ll figure out how many five-card hands have four aces. Since there are only four aces in the deck, there’s only one way to get all four of them in your hand. At that point there are 48 cards left for the other card in the hand. Using the fundamental counting principle, there are 1 x 48 ways to be dealt four aces.

EXAMPLE 2 Using Combinations to Compute Probability SOLUTION So there are 48 ways to get a five-card hand with four aces. The total number of hands with any 5 cards is the combinations of 5 cards chosen from 52, or 52C5. The probability of getting four aces is

EXAMPLE 3 Using Permutations to Compute Probability A combination lock has 40 numbers on it, from zero to 39. Find the probability that if the combination to unlock it consists of three numbers, it will contain the numbers 1, 2, and 3 in some order. Assume that numbers cannot be repeated in the combination. SOLUTION This is a permutation since the order of the numbers is important when you are unlocking the lock. The number of combinations for the lock containing 1, 2, and 3 is 3P3.

EXAMPLE 3 Using Permutations to Compute Probability SOLUTION There are 6 ways to create a combination with 1, 2, and 3. The total number of combinations is a permutation of the 40 numbers taken 3 at a time, or 40P3. The probability of the combination containing 1, 2, and 3 is

EXAMPLE 4 Using Combinations to Compute Probability A store has six different fitness magazines and three different news magazines. If a customer buys three magazines at random, find the probability that the customer will pick two fitness magazines and one news magazine. SOLUTION There are 6C2 or 15 ways to select two fitness magazines from six fitness magazines, as shown:

EXAMPLE 4 Using Combinations to Compute Probability SOLUTION There are 3C1 or three ways to select one magazine from three news magazines: Using the fundamental counting principle, there are 15 x 3 or 45 ways to select two fitness magazines and one news magazine.

EXAMPLE 4 Using Combinations to Compute Probability SOLUTION Next, there are 9C3 or 84 ways to select three magazines from nine magazines: The probability of selecting two fitness magazines and one news magazine is