Question

Illinois license plates used to consist of either three letters followed by three digits or two letters followed by four digits. Assume that all the different possible license plates are equally likely. (a) What is the probability that a randomly chosen plate contains the number 7777? (Round your answer to six decimal places.) (b) What is the probability that a randomly chosen plate contains the substring ME? (For example, ME4321 or PME786 are two ways ME might appear. Round your answer to six decimal places.)

Answer 1

Answer: (a) 0.000028 (b) 0.001479

Explanation:

Given : Illinois license plates used to consist of either three letters followed by three digits or two letters followed by four digits.

Number of ways to form a licence plate having three letters followed by three digits :-

`(26)^3*(10)^3=17576000`

Number of ways to form a licence plate having two letters followed by four digits :-

`(26)^2*(10)^4=6760000`

Total number of ways to form a licence plate : 17576000+6760000=24336000

(a) If a license place consists of 7777, then it must belongs to "two letters followed by four digits."

Then, the number of ways to make a license plate having number 7777 will be :_

`26*26*1^2=676`

Now, the probability that a randomly chosen plate contains the number 7777 :_

`=(676)/(24336000)=0.00002777777\approx0.000028`

(b) Consider ME as 1 thing.

The number of ways to make a license plate contains the substring ME will be :_

`(26)*1* (10)^3+(1)*(10)^4=26000+10000=36000`

Now, the probability that a randomly chosen plate contains the substring ME :-

`=(36000)/(24336000)=0.001479`