Question

Consider a bag that contains 220 coins of which 6 are rare Indian pennies. For the given pair of events A and​ B, complete parts​ (a) and​ (b) below. ​A: When one of the 220 coins is randomly​ selected, it is one of the 6 Indian pennies. ​B: When another one of the 220 coins is randomly selected​ (with replacement), it is also one of the 6 Indian pennies. a. Determine whether events A and B are independent or dependent. b. Find​ P(A and​ B), the probability that events A and B both occur.

Answer 1

a. The two events are dependent.

b.
`P(A\cap B)`=
`(1)/(220)`.

Explanation:

Given

Total coins =220

Number of Indian pennies= 6

A: When one of the 220 coins is randomly selected, it is one of the Indian pennies.

Therefore , the probability of getting an Indian pennies=
`(6)/(220 )`

By using formula of probability=
`(Number \; of\; favourable\; cases)/(total\; number \; of \;cases)`

Probability of getting an Indian pennies=
`(3)/(110)`

B: When another one of the 220 coins is randomly selected( with replacement) , It is also one of the Indian pennies.

Therefore, probability of getting an Indian pennies=
`(6)/(220)`

Probability of getting an Indian pennies =
`(3)/(110)`

`A\cap B`: 1

`P(A\cap B)=(1)/(220)`

If two events are independent. Then

`P(A\cap B)= P(A)* p(B)`

P(A).P(B)=
`(3)/(110) * (3)/(110)`=
`(9)/(12100)`

Hence,
`P(A\cap B)\\eq P(A).P(B)`

Therefore, the two events are dependent.

b. Probability that events A and B both occur

Number of favourable cases when both events A and B occur=1

Total coins=220

Probability=
`(Number \; of\; favourable \; cases)/(Total\; number\; of\; cases)`

`P(A\cap B)=(1)/(220)`