Question

A middle school principal has 80 keys on her key chain to distribute to staff on the first day of school. Of these 80 keys, 60 open classroom doors, 40 open the door to the teachers' lounge, and 30 open classroom doors and the teachers' lounge. What is P(C or T), the probability that a key opens a classroom or the teachers' lounge?

a. 0.25
b. 0.50
c. 0.75
d. 1.00

Answer 1

The probability that a key opens a classroom or the teachers' lounge is 0.125.

Step-by-step explanation:

To find the probability that a key opens a classroom or the teachers' lounge, we need to calculate P(C or T), which represents the probability of event C (opening a classroom) or event T (opening the teachers' lounge).

From the given information, we know that 30 keys open both a classroom and the teachers' lounge. This means that out of the 80 keys, (60 - 30) = 30 keys open only a classroom, and (40 - 30) = 10 keys open only the teachers' lounge.

To calculate the probability P(C or T), we add the probabilities of event C and event T, and subtract the probability of both events occurring simultaneously (C and T):

P(C or T) = P(C) + P(T) - P(C and T)

Probability of opening a classroom (P(C)) = 30/80 = 0.375

Probability of opening the teachers' lounge (P(T)) = 10/80 = 0.125

Probability of both opening a classroom and the teachers' lounge (P(C and T)) = 30/80 = 0.375

Using the formula for P(C or T), we have:

P(C or T) = 0.375 + 0.125 - 0.375 = 0.125

Therefore, the probability that a key opens a classroom or the teachers' lounge is 0.125.