# Probability

1, At Acme Inc., women employees are three times as likely to take advantage of computer training courses as men employees. Seventy percent of Acme's very large work force are women.

a, What proportion of Acme employees who take advantage of computer training courses are women?

b, Twelve percent of the workforce at Acme take computer training courses. What percent of the men employees take computer training courses?

2, a, At a Smith family reunion there are 28 John Smith's and 18 Jim Smiths in a group photograph. Five of those in the photograph are selected at random. Find the probability that the Johns outnumber the Jims amongst the five chosen.

b, Every time Jim Smith takes his car to the car wash there is a 24% probability that he will forget to retract the radio aerial. Next month he will take his car to the car wash nine times. What is the probability that he will forget to retract the aerial at most four times?

c, What is the mean number of times he will forget to retract, and the standard deviation in the number of times?

3, A time-and-motion-study consultant has been hired at E.I Ltd. She has indentified a certain work station as a bottleneck in production. Initial data suggest that the times required for processing pieces at this station may be treated as having an exponential distribution with a mean of 300 seconds.

a, If 50 pieces are processed, about how many of them take between 240 and 360 seconds to process?

The formula for the cumulative distribution function of the exponential distribution is

b, What is the probability that more than the mean period of pieces will be processed in a ten-minute period?

c, What is the median processing time at the station?

4, The fire department in a city holds fire frills in downtown office buildings. Records show that the times required to evacuate ten-storey buildings during fire drills are approximately normally distributed with a mean of 9 minutes and a standard deviation of 2 minutes. In a one month period 6 randomly selected ten-storey buildings held fire drills.

a, About how many of the buildings were evacuated in less than 8 minutes that month?

B, The fire department wishes to specify a length of time such that ninety-nine percent of ten-storey buildings will be evacuated in that amount of time or less. What length of time should be specified?

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#### Solution Preview

1, At Acme Inc., women employees are three times as likely to take advantage of computer training courses as men employees. Seventy percent of Acme's very large work force are women.

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<br><br>a, What proportion of Acme employees who take advantage of computer training courses are women?

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<br><br>Since women employees are three times as likely to take advantage of computer training courses as men employees, this means that among those who take advantage ratio of women to men is 3:1. Then the women proportion is 3/(3+1) = 0.75

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<br><br>b, Twelve percent of the workforce at Acme take computer training courses. What percent of the men employees take computer training courses?

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<br><br>From a we know that 0.75 of Acme employees who take advantage are women, then the rest 0.25 of those who take advantage are men, while the proportion of taking advantage of the workforce is 12%. So in the total workforce, the male proportion who take advantage is 0.25 * 12%=3%

<br><br>From the condition, we know, 70% of workforce are women, and 30% are men. Therefore the percentage of the men employees take courses is 3% / 30% = 0.1

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<br><br>2, a, At a Smith family reunion there are 28 John Smith's and 18 Jim Smiths in a group photograph. Five of those in the photograph are selected at random. Find the probability that the Johns outnumber the Jims amongst the five chosen.

<br><br>

<br><br>In this case, every photo chosen has two mutually exclusive possible outcomes (to John or Jim), then the binomial distribution can be used to determine the probability of obtaining exactly r successes in the N outcomes. The binomial probability for obtaining r successes in N ...

#### Solution Summary

1, At Acme Inc., women employees are three times as likely to take advantage of computer training courses as men employees. Seventy percent of Acme's very large work force are women.

a, What proportion of Acme employees who take advantage of computer training courses are women?

b, Twelve percent of the workforce at Acme take computer training courses. What percent of the men employees take computer training courses?

2, a, At a Smith family reunion there are 28 John Smith's and 18 Jim Smiths in a group photograph. Five of those in the photograph are selected at random. Find the probability that the Johns outnumber the Jims amongst the five chosen.

b, Every time Jim Smith takes his car to the car wash there is a 24% probability that he will forget to retract the radio aerial. Next month he will take his car to the car wash nine times. What is the probability that he will forget to retract the aerial at most four times?

c, What is the mean number of times he will forget to retract, and the standard deviation in the number of times?

3, A time-and-motion-study consultant has been hired at E.I Ltd. She has indentified a certain work station as a bottleneck in production. Initial data suggest that the times required for processing pieces at this station may be treated as having an exponential distribution with a mean of 300 seconds.

a, If 50 pieces are processed, about how many of them take between 240 and 360 seconds to process?

The formula for the cumulative distribution function of the exponential distribution is

b, What is the probability that more than the mean period of pieces will be processed in a ten-minute period?

c, What is the median processing time at the station?

4, The fire department in a city holds fire frills in downtown office buildings. Records show that the times required to evacuate ten-storey buildings during fire drills are approximately normally distributed with a mean of 9 minutes and a standard deviation of 2 minutes. In a one month period 6 randomly selected ten-storey buildings held fire drills.

a, About how many of the buildings were evacuated in less than 8 minutes that month?

B, The fire department wishes to specify a length of time such that ninety-nine percent of ten-storey buildings will be evacuated in that amount of time or less. What length of time should be specified?