Question

a number is chosen at random from S={1,...,520}. what is th probability that it is divisible by 5 or 7?

Answer 1

S={1,...,520}

To find the required probability 1st we need to find out the how many number in set S is divisible by 5,7 and by both 5 & 7.

To find how many number is divisible by 5 :

The first number divisible by 5 is 5

as 520/5 = 104 so last number divisible by 5 is 520

5,10,15,20,......,520 are the numbers divisible by 5 in S={1,...,520}

Number that are divisible by 5 in S are in A.P(Arithmetic Progression) series so we can use the A.P formula for last term to know how many number is divisible by 5 in S

a=5, d=5

Tn=520 [Last number divisible by 5]

Formula: Tn=a+(n-1)*d

520=5+(n-1)*5

520-5=(n-1)*5

n-1=515/5

n-1=103

n=104

Therefore,

probability that a random number is divisible by 5 in S = (favourable outcome /Total outcome) =(104/520) = 0.2

To find how many number is divisible by 7:

The first number divisible by 7 is 7

as 520/7=74.28 so last number divisible by 7 is = ( 74*7) = 518

7,14,21,28,......,518 are the numbers divisible by 7 in S={1,...,520}

It is also an A.P so using the same formula as earlier

a=7, d=7,Tn=518

Formula: Tn=a+(n-1)*d

518=7+(n-1)*7

518-7=(n-1)*7

n-1=511/7

n-1=73

n=74

Therefore,

probability that a random number is divisible by 7 in S = (favourable outcome /Total outcome) =(74/520) = 0.1423

To find the numbers that is divisible by 5 and 7 we have to find the numbers in S that is divisible by 35 (as 5*7=35)

To find how many number is divisible by 35:

The first number divisible by 35 is 35

as 520/35=14.86 so last number divisible by 35 is = ( 14*35) = 490

35,70,105,140,......,490 are the numbers divisible by 35 in S={1,...,520}

It is also an A.P

a=35, d=35,Tn=490

Formula: Tn=a+(n-1)*d

490=35+(n-1)*35

490-35=(n-1)*35

n-1=455/35

n-1=13

n=14

Therefore,

probability that a random number is divisible by 5 and 7 in S = (favourable outcome /Total outcome) =(14/520) = 0.0269

Let :

P(A) = probability that a random number is divisible by 5 in S = 0.2

P(B) = probability that a random number is divisible by 7 in S = 0.1423

then P(A B) = probability that a random number is divisible by 5 and 7 in S = 0.0269

So, P(A B) = Probability that a random number is divisible by 5 or 7 in S

P(A B) = P(A) + P(B) - P(A B)

=0.2 + 0.1423 - 0.0269 = 0.3154

Answer: Probability that a number chosen randomly from S={1,......,520} is divisible by 5 or 7 is 0.3154