Bob sets two alarm clocks (battery-powered) to be sure he arises for his Monday 8:00 A.M. accounting
exam. There is a 75 percent chance that either clock will wake Bob. (a) What is the probability
that Bob will oversleep? (b) If Bob had three clocks, would he have a 99 percent chance of
waking up?
no
To calculate the probability that Bob will oversleep, we need to calculate the probability of both alarm clocks failing to wake him up.
(a) Probability that Bob will oversleep with two alarm clocks:
The probability that the first alarm clock doesn't wake Bob up is 25% (1 - 0.75), and the probability that the second alarm clock doesn't wake him up is also 25%. Since the two alarm clocks are independent events, we can multiply these probabilities to get the probability of both clocks failing:
P(oversleeping) = P(first clock fails) × P(second clock fails)
= 0.25 × 0.25
= 0.0625
So, there is a 6.25% chance that Bob will oversleep.
(b) Now, let's consider if Bob had three alarm clocks:
Assuming that each clock has a 75% chance of waking him up, we can calculate the probability of all three alarm clocks failing to wake him up using the same approach as in (a):
P(oversleeping) = P(first clock fails) × P(second clock fails) × P(third clock fails)
= 0.25 × 0.25 × 0.25
= 0.015625
So, there is a 1.56% chance that Bob will oversleep if he had three alarm clocks.
Therefore, even with three alarm clocks, Bob would not have a 99% chance of waking up.
(a) To find the probability that Bob will oversleep, we need to find the probability that both alarm clocks fail to wake him up.
The probability that the first alarm clock fails to wake Bob up is 1 - 0.75 = 0.25 (since there is a 75% chance it will wake him up).
Similarly, the probability that the second alarm clock fails to wake Bob up is also 0.25.
To find the probability that both alarm clocks fail, we multiply the probabilities together:
P(both fail) = P(first fails) * P(second fails) = 0.25 * 0.25 = 0.0625
Therefore, the probability that Bob will oversleep is 0.0625, or 6.25%.
(b) If Bob had three clocks, we can use a similar approach to find the probability of oversleeping.
The probability that a single alarm clock fails to wake Bob up is still 0.25 (since each clock has a 75% chance of waking him up).
To find the probability that all three alarm clocks fail, we multiply the probabilities together:
P(oversleeping with three clocks) = P(first fails) * P(second fails) * P(third fails) = 0.25 * 0.25 * 0.25 = 0.015625
Therefore, the probability that Bob will oversleep with three clocks is 0.015625, or 1.5625%.