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 2222? (Round your answer to six decimal places.)


(b) What is the probability that a randomly chosen plate contains the substring HI? (For example, HI4321 or PHI786 are two ways HI might appear. Round your answer to six decimal places.)

Answer 1

The probability that a randomly chosen plate contains the number 2222 is 0.000028 approximately.

The probability that a randomly chosen plate contains the sub-string HI is 0.002548 approximately.

Explanation:

Consider the provided information.

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

Part (A)

Let A is the ways in which plates consist of three letters followed by three digits and B is the ways in which two letters followed by four digits.

Here repetition is allow. The number of alphabets are 26 and the number of distinct digits are 10.

The numbers of ways in which three letters followed by three digits can be chosen is:
`26* 26* 26 * 10 * 10 *10`

`26^3* 10^3=17576000`

The numbers of ways in which two letters followed by four digits can be chosen is:
`26* 26* 10 * 10 * 10 *10`

`26^2* 10^4=6760000`

Hence, the total number of ways are 17576000 + 6760000 = 24336000

Randomly chosen plate contains the number 2222 that means the first two letter can be any alphabets but the rest of the digit should be 2222.

Thus, the total number of ways that a randomly chosen plate contains the number 2222 number are:
`26^2=676`

The probability that a randomly chosen plate contains the number 2222 is:

`(676)/(24336000) \approx 0.000028`

Part (B)

The number of ways in which chosen plate contains the sub-string HI:

If three letters followed by three digits plate contains the sub-string HI, then the number of possible ways are:

`26* 1*10^3+1* 26*10^3`

If two letters followed by four digits plate contains the sub-string HI, then the number of possible ways are:

`1* 10^4`

Thus, the total number of ways that a randomly chosen plate contains the sub-string HI are:

`1* 10^4+26* 1*10^3+1* 26*10^3`

`62000`

From part (A) we know that the total number of ways to chose a number plate is 24336000.

The probability that a randomly chosen plate contains the sub-string HI is:

`(62000)/(24336000) \approx 0.002548`