Let's break down each of these probability problems using the extended multiplication rule, which states that the probability of a series of independent events happening is the product of the individual probabilities.

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(a) Probability of getting 5 heads when flipping a fair coin 5 times.

• A fair coin has two outcomes: heads (H) and tails (T), each with a probability of $\frac{1}{2}$.
• The flips are independent, so we multiply the probabilities for each flip.

The probability of getting heads on each flip is $\frac{1}{2}$. For 5 heads in a row, the probability is: $P(\text{5 heads}) = \left(\frac{1}{2}\right)^5 = \frac{1}{32} = 0.03125$ Rounded to 3 digits:
Blank 1 = 0.031

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(b) Probability that 3 randomly selected people were born on the same day of the week.

• There are 7 days in a week, so the probability that the first person was born on any particular day is 1 (it's certain).
• The second person must be born on the same day, so the probability for them is $\frac{1}{7}$.
• The third person must also be born on that same day, so their probability is also $\frac{1}{7}$.

The total probability is: $P(\text{same day}) = 1 \times \frac{1}{7} \times \frac{1}{7} = \frac{1}{49} \approx 0.0204$ Rounded to 3 digits:
Blank 2 = 0.020

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(c) Probability that the car will start for 14 days in a row, given that the car starts 99% of the time.

• The probability of the car starting on any given day is $0.99$.
• Since the events are independent, we multiply the probability for each of the 14 days.

The total probability is: $P(\text{car starts 14 days}) = 0.99^{14} \approx 0.868$ Rounded to 3 digits:
Blank 3 = 0.868

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Summary of answers:

• Blank 1 = 0.031
• Blank 2 = 0.020
• Blank 3 = 0.868

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Would you like more detailed explanations or have any questions about the calculations?

Further Related Questions:

1. What would the probability be if you flipped the coin 10 times and wanted 10 heads?
2. What is the probability that at least 2 people out of 3 were born on the same day of the week?
3. What is the probability that your car will fail to start at least once in 14 days?
4. How would the probabilities change if you were selecting 5 people instead of 3?
5. How do you calculate the probability for events that are not independent?

Tip: When dealing with repeated independent events, remember that the extended multiplication rule is your best tool to calculate the probability for all events happening together.