Math

QuestionA student's alarm clock has a 7.5%7.5\% daily failure rate. What is the probability the clock fails on the day of an important exam? If the student has two clocks, what is the probability both fail?

Studdy Solution

STEP 1

Assumptions
1. The failure rate of the student's alarm clock is 7.5% daily.
2. The probability that the alarm clock will not work on any given morning is independent of any other morning.
3. For part (b), the student has two identical alarm clocks with the same failure rate, and their failures are independent events.

STEP 2

For part (a), we need to find the probability that the alarm clock will not work on the morning of an important final exam. The probability of failure is already given as 7.5%.
P(Alarm clock fails)=7.5%P(\text{Alarm clock fails}) = 7.5\%

STEP 3

Convert the percentage to a decimal to facilitate calculations.
7.5%=0.0757.5\% = 0.075

STEP 4

The probability that the alarm clock will not work on the morning of an important final exam is the same as the daily failure rate.
P(Alarm clock fails on exam morning)=0.075P(\text{Alarm clock fails on exam morning}) = 0.075
This is the answer to part (a).

STEP 5

For part (b), we need to find the probability that both alarm clocks fail on the morning of an important final exam. Since the failures of the two alarm clocks are independent events, we can multiply their individual failure probabilities to find the joint probability of both failing.
P(Both alarm clocks fail)=P(Alarm clock 1 fails)×P(Alarm clock 2 fails)P(\text{Both alarm clocks fail}) = P(\text{Alarm clock 1 fails}) \times P(\text{Alarm clock 2 fails})

STEP 6

Since both alarm clocks have the same failure rate, we use the failure rate from part (a) for both clocks.
P(Both alarm clocks fail)=0.075×0.075P(\text{Both alarm clocks fail}) = 0.075 \times 0.075

STEP 7

Calculate the probability that both alarm clocks fail.
P(Both alarm clocks fail)=0.075×0.075=0.005625P(\text{Both alarm clocks fail}) = 0.075 \times 0.075 = 0.005625

STEP 8

Round the result to five decimal places as instructed.
P(Both alarm clocks fail)0.00563P(\text{Both alarm clocks fail}) \approx 0.00563
This is the answer to part (b).

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord