Answer: (a) 0.000028 (b) 0.001479
Step-by-step 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 :-
[tex](26)^3\times(10)^3=17576000[/tex]
Number of ways to form a licence plate having two letters followed by four digits :-
[tex](26)^2\times(10)^4=6760000[/tex]
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 :_
[tex]26\times26\times1^2=676[/tex]
Now, the probability that a randomly chosen plate contains the number 7777 :_
[tex]=\dfrac{676}{24336000}=0.00002777777\approx0.000028[/tex]
(b) Consider ME as 1 thing.
The number of ways to make a license plate contains the substring ME will be :_
[tex](26)\times1\times (10)^3+(1)\times(10)^4=26000+10000=36000[/tex]
Now, the probability that a randomly chosen plate contains the substring ME :-
[tex]=\dfrac{36000}{24336000}=0.001479[/tex]
