A consumer organization estimates that over a​ 1-year period 1616​% of cars will need to be repaired​ once, 55​% will need repairs​ twice, and 44​% will require three or more repairs. If you own two​ cars, what is the probability that​a) neither will need​ repair?​b) both will need​ repair?​c) at least one car will need​ repair? Please show the work on how to calculate the answer to the above. I have several questions similar to this one in my homework. Thank you, Linda

Question
Asked Feb 26, 2019
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A consumer organization estimates that over a 1-year period
1616%
of cars will need to be repaired once,
55%
will need repairs twice, and
44%
will require three or more repairs. If you own two cars, what is the probability that
a) neither will need repair?
b) both will need repair?
c) at least one car will need repair?
 
Please show the work on how to calculate the answer to the above. I have several questions similar to this one in my homework. Thank you, Linda
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Expert Answer

Step 1

Given information:

From the estimation of car consumer organization for a period of 1 year, the following things can be noted:

Let A be the event that car needs repair once.

The probability that car needs repair once is P(A) = 0.16.

Let B be the event that car needs repair twice.

The probability that car needs repair twice is P(B) = 0.05.

Let C be the event that car needs repair more than twice.

The probability that car needs repair more than twice is P(C) = 0.04

Let D be the event that car does not need any repairs.

The probability that car does not need any repairs is P(D) = 1 – 0.16 – 0.05 –0.04 = 0.75.

Here, any two of the events A, B,C and D will not occur at the same time. That is, a car that comes under the category of one repair does not come under the category of 2 repairs, 3 or more repairs and 0 repair.

Therefore, the intersection of any of the two events is zero.

Hence, the events A, B,C and D are mutually exclusive.

Step 2

a.

Find the probability that none of the two cars owned by investigator needs repair:

Here, it is given that the investigator has own two cars.

The requirement is both the cars does not need repair.

The probability that none of the two cars owned by investigator needs repair is obtained as 0.5625 from the calculation given below:

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Step 3

b.

Find the probability that both the two cars owned by investigator needs repair:

The requirement is both the cars need repair.

Here, the repair might be once, twice or more than twice. That is, the category of a car needs repair contains the categori...

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