Current learning objective: Calculations of Probabilities Question 24 Score: 0 of 4 points We flip a coin 25 times \( (n=25) \) and note that it surprisingly comes up heads \( 90 \%(p=0.9) \) of the time. The mean and standard deviation for the number of times the coin lands on heads is \( \mu=22.5 \) and \( \sigma=1.5 \). Assume the number of times the coin lands on heads is normally distributed. Solve the following: a. The 16 th and 84 th percentile of the number of times the coin lands on heads are Enter your answer answers to the nearest whole number) b. There is about a Enter your answer c. There is about a Enter your answer

To find the 16th and 84th percentiles in a normal distribution, we can use the mean and standard deviation provided. The 16th percentile corresponds to approximately -1 standard deviation from the mean, which gives us: \[ P(16) \approx \mu - \sigma = 22.5 - 1.5 = 21 \] The 84th percentile corresponds to approximately +1 standard deviation from the mean: \[ P(84) \approx \mu + \sigma = 22.5 + 1.5 = 24 \] So, the answers are 21 and 24 for parts (a). For part (b), the probability that the number of heads is less than 21 (the 16th percentile) is about 16%, and for part (c), the probability of getting more than 24 heads (the 84th percentile) is also about 16%. Thus, one could say that there is about a 16% chance the coin will land on heads less than 21 times, and a 16% chance it will land on heads more than 24 times. Clearly, flipping a coin is more fun than doing math, but the probability aspects ensure we have a fair understanding of randomness! Happy flipping!