Section 12.6 OR and AND Problems

What You Will Learn Compound Probability OR Problems AND Problems Independent Events

Compound Probability In this section, we learn how to solve compound probability problems that contain the words and or or without constructing a sample space.

OR Probability The or probability problem requires obtaining a “successful” outcome for at least one of the given events.

Probability of A or B To determine the probability of A or B, use the following formula.

Example 1: Using the Addition Formula Each of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 is written on a separate piece of paper. The 10 pieces of paper are then placed in a hat, and one piece is randomly selected. Determine the probability that the piece of paper selected contains an even number or a number greater than 6.

Example 1: Using the Addition Formula Solution Draw a Venn Diagram

Example 1: Using the Addition Formula Solution The seven numbers that are even or greater than 6 are 2, 4, 6, 7, 8, 9, and 10.

Mutually Exclusive Two events A and B are mutually exclusive if it is impossible for both events to occur simultaneously. If two events are mutually exclusive, then the P(A and B) = 0. The addition formula simplifies to

Example 3: Probability of A or B One card is selected from a standard deck of playing cards. Determine whether the following pairs of events are mutually exclusive and determine P (A or B). a) A = an ace, B = a 9 Solution Impossible to select both so

Example 3: Probability of A or B b) A = an ace, B = a heart Solution Possible to select the ace of hearts, so NOT mutually exclusive

Example 3: Probability of A or B c) A = a red card, B = a black card Solution Impossible to select both so mutually exclusive

Example 3: Probability of A or B d) A = a picture card, B = a red card Solution Possible to select a red picture card, so NOT mutually exclusive

And Problems The and probability problem requires obtaining a favorable outcome in each of the given events.

Probability of A and B To determine the probability of A and B, use the following formula.

Probability of A and B Since we multiply to find P (A and B), this formula is sometimes referred to as the multiplication formula. When using the multiplication formula, we always assume that event A has occurred when calculating P(B) because we are determining the probability of obtaining a favorable outcome in both of the given events.

Example 5: An Experiment without Replacement Two cards are to be selected without replacement from a deck of cards. Determine the probability that two spades will be selected.

Example 5: An Experiment without Replacement Solution The probability of selecting a spade on the first draw is 13/52. Assuming we selected a spade on the first draw, then the probability of selecting a spade on the second draw is 12/51.

Example 5: An Experiment without Replacement Solution

Independent Events Event A and event B are independent events if the occurrence of either event in no way affects the probability of occurrence of the other event. Rolling dice and tossing coins are examples of independent events.

Example 6: Independent or Dependent Events? One hundred people attended a charity benefit to raise money for cancer research. Three people in attendance will be selected at random without replacement, and each will be awarded one door prize. Are the events of selecting the three people who will be awarded the door prize independent or dependent events?

Example 6: Independent or Dependent Events? Solution The events are dependent since each time one person is selected, it changes the probability of the next person being selected. P(person A is selected) = 1/100 If person B is actually selected, then on the second drawing, P(person A is selected) = 1/99

Independent or Dependent Events? In general, in any experiment in which two or more items are selected without replacement, the events will be dependent.