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Chapter 5: Problem 89

Suppose all the days of the week are equally likely as birthdays. Alicia and David are two randomly selected, unrelated people. a. What is the probability that they were both born on Monday? b. What is the probability that Alicia OR David was born on Monday?

Short Answer

Expert verified
a. The probability that both Alicia and David were born on Monday is \( \frac{1}{49} \). b. The probability that either Alicia or David was born on Monday is \( \frac{13}{49} \).

Step by step solution

01

Calculate the Probability of Both Born on Monday

First, we calculate the probability that Alicia and David were both born on Monday. Since their birthdays are independent events, the probability that both were born on Monday is the product of their individual probabilities. Each individual has a 1 in 7 chance of being born on a Monday since there are 7 days in a week. Hence, by multiplying these probabilities, \( \frac{1}{7} \times \frac{1}{7} = \frac{1}{49} \) is the probability that both were born on Monday.
02

Calculate the Probability of Either Born on Monday

Next, we calculate the probability that either Alicia or David was born on Monday. For this, we use the principle of sum of probabilities since these are mutually exclusive events. The probability of either Alicia or David being born on Monday would be the sum of their individual probabilities. But, this includes also the case when they both have birthday on Monday, which is counted twice in this sum. Hence, the sum of the probabilities minus the probability both have birthday on Monday is the correct probability. Hence, \( 2 \times \frac{1}{7} - \frac{1}{49} = \frac{13}{49} \) is the probability that either Alicia or David was born on Monday.

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