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Practice Problems in Probability and Statistics

Question 1

Question 1

The table below depicted the results of a survey conducted by a shopping mall after the Christmas Sales. Have you bought anything? Male Female Total

Yes 20 23 43

No 15 42 57

Total 35 65 100

Suppose a person is randomly selected from the sample.

( a ) Find the probability that the selected person is a male.

( b ) Find the probability that the selected person bought something.

( c ) Find the probability that the selected person is a male and has bought something in the sales.

( d ) Find the probability that the selected person is a male and he bought nothing in the sales.

( e ) Find the probability that the selected person bought something given that the person is a male.

( f ) Find the probability that the selected person bought something given that the person is a female.

( g ) Determine whether the purchasing tendency is independent with the gender.

Question 2

In the following table, a sample of university students are classified according to their class standing and also according to their most favorite pizza topping:  Anchovies (A) Onions (O) Mushrooms (M) Hamburger (H) Freshman (F) 7 6 7 3 Sophomore (S) 1 9 0 9 Junior (J) 3 2 5 8 If one of these students is selected at random (Let F, S and J denote the three classes, and let A, O, M and H denote the four pizza toppings).

Question 3

The members of a consulting firm rent cars from three rental agencies: 52% from agency A, 36% from agency B and 12% from agency C. From previous statistics, it is known that 7% of the cars from agency A need a tune-up, 22% of the cars from agency B need a tune-up, and 5% of the cars from agency C need a tune-up. ( a ) Construct a tree diagram for the above scenario.

( b ) Compute the probability that:

( i ) the rental car delivered to the firm will need a tune-up;

( ii ) the rental car delivered comes from agency B and need a tune-up.

( c ) Given that a car does not need a tune-up, what is the probability that the car comes from agency B.

Question 4

Suppose in the production line, 10% of the light bulbs are defective. A customer complained on the poor quality of the light bulbs and asked for a checking. A random sample of 20 light bulbs is selected and sent for inspection.

( a ) What is the probability that there is exactly one defective light bulb in the sample?
( b ) What is the probability that there are at least 3 defective light bulbs in the sample?
( c ) Find the mean and the standard deviation of the number of defective light bulb in the sample.

Question 5

The number of traffic accidents follows a Poisson distribution with mean 3 (per month).
( a ) What is the probability that in any given month at this intersection

( i ) exactly 5 accidents will occur?
( ii ) less than 3 accidents will occur?
( iii ) at least 2 accidents will occur?

( b ) What is the probability that there are exactly 4 accidents in a two-month period?

Question 6

Suppose the time it takes a technician to fix a computer system follows an exponential distribution with a mean of 45 minutes.

( a ) What is the probability of fixing a computer system in 50 minutes or less?

( b ) What is the probability of fixing a computer system between 40 and 50 minutes?