Question

Can someone explain how to answer these 3 math problems:

1. If 6 fair coins are flipped, what is the probability that at least one of the coins will land with tails facing up?

2. A person is rolling a fair, six-sided die until they roll a 5. What is the probability that it takes them at least two

attempts to roll their first 5?

3. During heavy rain, a basement’s three pumps (pump A, pump B, and pump C) must all function correctly, or the

basement will flood. If the pumps’ probabilities of working are 33%, 60% and 86% respectively, what is the probability

that the basement will flood? (Assume the pumps work independently)

Answer 1

1.

The probability that at least one of 6 fair coins will land with tails facing up is 1 - (the probability that all 6 coins will land heads up).

The probability that a single coin will land heads up is 1/2, so the probability that all 6 coins will land heads up is (1/2)^6 = 1/64.

Therefore, the probability that at least one coin will land tails up is 1 - (1/64) = 63/64.

2.

The probability that it takes a person at least two attempts to roll their first 5 is 1 - (the probability that they roll a 5 on their first attempt).

The probability that a single roll of a die will result in a 5 is 1/6, so the probability that a person will roll a 5 on their first attempt is 1/6. Therefore, the probability that it takes them at least two attempts to roll their first 5 is 1 - (1/6) = 5/6.

3.

The probability that the basement will flood is 1 - (the probability that all 3 pumps will work).

The probability that pump A will work is 33%, the probability that pump B will work is 60%, and the probability that pump C will work is 86%. The probability that all 3 pumps will work is (33%)(60%)(86%) = 1629/2160. Therefore, the probability that the basement will flood is 1 - (1629/2160) = 59/240.

A detailed explanation of how to calculate the probability that the basement will flood:

• The probability that pumps A will work is 33%.
• The probability that pump B will work is 60%.
• The probability that pump C will work is 86%.
• The probability that all 3 pumps will work is (33%)(60%)(86%) = 1629/2160.
• The probability that at least one pump will fail is 1 - (the probability that all 3 pumps will work) = 1 - 1629/2160 = 531/2160.

Therefore, the probability that the basement will flood is 59/240.

Please let me know if you have any other questions.