Question

a recent survey asked 150 mathematics professors about how they invest their money between stocks, bonds, and certificates of deposit. the results of the survey are shown. complete parts (a) and (b). out of the 150 mathematics professors surveyed, the survey revealed that: 111 invested in stocks; 101 invested in bonds; 97 invested in certificates of deposit; 80 invested in stocks and bonds; 83 invested in bonds and certificates of deposit; 82 invested in stocks and certificates of deposit; 5 did not invest in any of the three. (a) what is the probability that a randomly chosen professor invested in stocks and bonds? the probability is \boxed{}. (type an integer or a simplified fraction.)

Explanation:

Step1: Determine the number of professors who invested in at least one of the three.

The total number of professors surveyed is 150, and 5 did not invest in any of the three. So the number of professors who invested in at least one is $150 - 5 = 145$.

Step2: Use the principle of inclusion - exclusion to find the number of professors who invested in stocks and bonds.

Wait, actually, the problem (a) directly gives the number of professors who invested in stocks and bonds as 80? Wait, no, wait. Wait, the question is the probability that a randomly chosen professor invested in stocks and bonds. The number of professors who invested in stocks and bonds is 80? Wait, no, wait, let's check again. Wait, the total number of professors who invested in at least one is 145, but the number of professors who invested in stocks and bonds is given as 80? Wait, no, the problem says "80 invested in stocks and bonds". Wait, but let's confirm. Wait, the total number of professors surveyed is 150, 5 did not invest in any, so 145 invested in at least one. But the number of professors who invested in stocks and bonds is 80. Wait, but the probability is the number of professors who invested in stocks and bonds divided by the total number of professors surveyed. Wait, the total number of professors is 150. So the number of professors who invested in stocks and bonds is 80? Wait, no, wait, maybe I misread. Wait, the problem says "80 invested in stocks and bonds". So the number of favorable outcomes (professors who invested in stocks and bonds) is 80, and the total number of possible outcomes (total professors) is 150. So the probability is $\frac{80}{150}$, which simplifies to $\frac{8}{15}$? Wait, no, wait, 80 divided by 150: divide numerator and denominator by 10, get $\frac{8}{15}$? Wait, 80 ÷ 10 = 8, 150 ÷ 10 = 15. Wait, but wait, is the number of professors who invested in stocks and bonds 80? Let's check the inclusion - exclusion principle to find the number of professors who invested in all three, but the question (a) is just the number who invested in stocks and bonds, which is given as 80? Wait, the problem states: "80 invested in stocks and bonds". So the number of professors who invested in stocks and bonds is 80. So the probability is the number of professors who invested in stocks and bonds divided by the total number of professors surveyed (150). So we have:

Number of professors who invested in stocks and bonds = 80

Total number of professors = 150

Probability = $\frac{80}{150}=\frac{8}{15}$ (simplifying by dividing numerator and denominator by 10)

Wait, but let's confirm. Wait, the total number of professors is 150, 5 did not invest in any, so 145 invested in at least one. But the number of professors who invested in stocks and bonds is 80, which is part of the 145. But the probability is based on the total number of professors, not the number who invested in at least one. Because we are choosing a random professor from all 150. So the number of professors who invested in stocks and bonds is 80, so the probability is $\frac{80}{150}=\frac{8}{15}$.

Wait, but let's check the calculation again. 80 divided by 150: GCD of 80 and 150 is 10. 80 ÷ 10 = 8, 150 ÷ 10 = 15. So $\frac{8}{15}$.

Answer:

$\frac{8}{15}$