A health club monitors the average heart rate of its members when using a cardiovascular workout machine. The lower quartile of the heart rate is 115, the upper quartile is 165. What is the probability that when a group of 4 members is chosen at random at least 3 of them had an average heart rate of more than 165 when doing this workout?

Answer:

8) C

Upper quartile means 75% to 100%.
Difference = 100 – 75 = 25%

Probability of a person having a heart beat of 165 or more = 25% ÷ 100% = ¼

4 members were chosen. 3 would have probability of ¼ and the fourth would have probability of ¾.
Possibilities of the order of people = YYYN, YYNY, YNYY, NYYY
Y = person has a heart beat of more than 165
N = person has a heart beat of less than 165

So there are 4 possibilities.

Probability of 3 members having heart beat 165 or more = ¼ x ¼ x ¼ x ¾ x 4 = ¹²⁄₂₅₆

The question says at least 3 people. So we can also obtain 4 people with a heart beat of more than 165.

Probability of this = ¼ x ¼ x ¼ x ¼ = ¹⁄₂₅₆

Probability of at least 3 people having a hear beat of more than 165 = ¹²⁄₂₅₆ + ¹⁄₂₅₆ = ¹³⁄₂₅₆