Question

The rental car agency has 30 cars on the lot. 10 are in great shape, 16 are in good shape, and 4 are in poor shape. Four cars are selected at random to be inspected. Do not simplify your answers. Leave in combinatorics form. What is the probability that:

a. Every car selected is in poor shape

b. At least two cars selected are in good shape.

c. Exactly three cars selected are in great shape.

d. Two cars selected are in great shape and two are in good shape.

e. One car selected is in good shape but the other 3 selected are in poor shape.

Answer 1

Solution :

Total number of cars = 30

Number of cars in great shape = 10

Number of cars in good shape = 16

Number of cars in poor shape = 4

4 car is selected.

Probability of an event E is given by,

P(E) = number of favourable outcomes/total number of outcomes

a) We have to obtain the probability that every car selected is in poor shape.

Number of ways of selecting 4 poor shaped cars from 4 poor shaped cars is given by,

Total number of ways of selecting 4 cars from total 30 cars is given by,

Hence, probability that every car selected is in poor shape is,

b) We have to obtain at least two cars selected are in good shape.

If at least two cars selected are in good shape, then it means either 2 cars are in good shape or 3 cars is in good shape or 4 cars is in good shape.

Case 1 : 2 cars selected are in good shape then other 2 cars selected will be from great and poor shape.

Total number of ways of selecting 2 cars from 16 good shape cars and 2 cars from remaining 14 cars is given by,

Case 2 : 3 cars selected are in good shape then other 1 selected car will be from great and poor shape.

Total number of ways of selecting 3 cars from 16 good shape cars and 1 cars from remaining 14 cars is given by,

Case 3 : All the 4 cars selected is in good shape.

Total number of ways of selecting 4 cars from 16 good shape cars is given by,

Hence, probability that at least two cars selected are in good shape is,

c) We have to obtain exactly three cars selected are in great shape.

If exactly three cars selected are in great shape, then other one selected will be from good and poor shape cars.

Total number of ways of selecting 3 cars from 10 great shape cars and 1 car from remaining 20 cars is given by,

Hence, probability that exactly three cars selected are in great shape is,

d) We have to find the probability that two cars selected are in great shape and two are in good shape.

Total number of ways of selecting 2 cars from 10 great shape cars and 2 cars from 16 good shape cars is given by,

Hence, probability that two cars selected are in great shape and two are in good shape is,

e) We have to obtain the probability that one car selected is in good shape but the other 3 selected are in poor shape.

Total number of ways of selecting 1 car from 16 good shape cars and 3 cars from 4 poor shape cars is given by,

Hence, probability that that one car selected is in good shape but the other 3 selected are in poor shape is,

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