# What is the probability that the fifth success will occur on either the tenth or eleventh trial ?

A society has as members 20 women, 18 men, and 30 children.

a) How many ways are there to select a committee of 3 women, 2 men, and 1 child?

5232600

b) How many ways are there to select a committee of 6 members if the only restriction is that there be at least 1 woman on the committee?

97181832

c) How many ways are there to select a committee of 6 members if the only restriction is that there be at least 1 woman and at least 1 man on the committee?

81884907

2. The probability of success for a given experiment is 0.36 (every time)

a) If there are twelve trials, what is the probability that the number of successes will be within one standard deviation of the expected number of successes ?

E(X) = 4.32 ?(X) = 1.66… P(3) + P(4) + P(5) = 0.63

b) What is the probability that the fifth success will occur before the ninth trial ? (try lateral thinking)

0.118

3. A new configuration of the computer key board has 10 of the 26 letters
of the alphabet in different positions from the standard keyboard, and
the rest are in the usual places. If a perfectly accurate touch typist types the word computer on the new keyboard without looking, what is the probability that there will be at least two mistakes in the resulting printed word ? It may be assumed that the typist does not know the new configuration.

Hypergeometric 0.9185

4. The probability of success for a given experiment is 0.6 (every time)

a) If there are twelve trials, what is the probability that there will be more than the mean number of successes ?

0.438

b) What is the probability that the fifth success will occur on either the tenth or eleventh trial ? (hard)

0.1672

5. In a group of 1875 scientists, 900 were educated in Canada, and the rest were educated elsewhere. Only 75 of the scientists were educated in Eurelia. A random selection was made of eight of the scientists.

a) What is the exact probability that at most four of them were educated in Canada ? Use an approximate distribution to get an approximation to the exact answer as well.

Exact (hypergeometric) 0.679883632….

Approximation (binomial) 0.6795

b) What is the exact probability that at most one of them was educated in Eurelia Use the approximate distribution that to get an approximation to the exact answer as well.

Exact (hypergeometric) 0.962191038

Approximation (binomial) 0.96185

6. a) The number of computer interruptions per day at a plant has (at least
approximately) a Poisson distribution. Last year, in the 366 days of
operation of the plant, there were only 10 days with no computer
interruptions. What was the average (mean) number of interruptions
per day ? How many interruptions were there in all during the year ?

b) A new configuration of the computer key board has 10 of the 26 letters
of the alphabet in different positions from the standard keyboard, and
the rest are in the usual places. If a perfectly accurate touch typist types the word computer on the new keyboard without looking, what is the probability that there will be at least two mistakes in the resulting printed word ? It may be assumed that the typist does not know the new configuration.

7. a) The number of arrivals of cement trucks at a construction site in an hour may be
treated as having a Poisson distribution with a mean of 4.6 per hour. What is the probability that more than 2 trucks will arrive in a randomly-selected half-hour time period?

b) Show that the distribution of the number of trucks arriving in an hour is positively skewed.

8. A printer prints 80 characters per line. The printer is old, and sometimes makes mistakes in printing characters. Each character has the same probability of being misprinted. A random sample of 100 lines produced by the printer only has 7 lines with no misprints. About how many of the 100 lines do you expect to have precisely 1 misprint amongst the 80 characters?

9. The traffic lights near the home of A.J.Jones change colour at random times, so that Jones considers the number of changes in any given time period to have a Poisson distribution, with a mean of 20 changes per hour.

a) What is the probability of more than the mean number of changes in a nine-minute period?

b) What is the median number of changes in a nine-minute period?

c) A.J. Jones tests his theory by counting the number of changes in a randomly-selected nine-minute period every day for many days. What is the probability that the first time that there are more than the mean number of changes is the fourth day that Jones checks?

c) What is the probability of having to wait more than 3 minutes for the light to change?

10. The number of flaws in 10-metre lengths of cable has approximately a
Poisson distribution with a mean of 0.24 flaws per metre.

a) What is the probability that twenty metres of the cable have at least 4 flaws?

b) A skeptical engineering group counted the number of flaws per 10-metre length for 400 lengths. They found that 32 of the 400 lengths had absolutely no flaws at all. What was the estimated figure that the engineering group obtained for the mean number of flaws per 10-metre length?

11. A group of 60 stocks includes 36 that have increased in value since November 1st, and 24 that have decreased since November 1st.

a) A.J.Jones has taken a random sample of 6 stocks from the group, and would like to calculate (before looking to see, of course) the probability of having picked at least 4 of the stocks that have increased in value. He cant remember whether to use the binomial or hypergeometric formula. Help him out by demonstrating that for three decimal places of accuracy he can use either one.