calculating this question for part b) please help. At a Business Fun Fair stall,

Question

calculating this question for part b) please help.

At a Business Fun Fair stall, players play a game to win 2 different prizes. It is known that the probabilities of John and Peter winning a game are 0.7 and 0.6 respectively. Each person will play the game 6 times independently. The person will win the 1st prize if he wins all 6 times and the 2nd prize if he wins exactly 5 times. What is the probability that

(a) John will win a prize?

(b) John wins the 2nd prize and Peter wins the 1st prize, given that each of them has won a prize?

Explanation / Answer

(b)

Accourding to Probability Distribution,

Probability of x success in n trials = n C x ( Probability of Success ) x   * ( Probability of Failure ) ( n - x )

For John to win 1st prize he has win all 6 times. So,

Probability of Success of John = 0.7

Probability of Failure of John = 1 - 0.7 = 0.3

Probability that John wins the 1st prize = 6 C 6 * ( 0.7 ) 6 * ( 0.3 ) 6

= ( 0.7 ) 6 * ( 0.3 ) 6

Similarly,

For John to win 2nd prize he has win 5 times out of 6 chances. So,

Probability that John wins the 2nd prize = 6 C 5* ( 0.7 ) 5 * ( 0.3 ) ( 6 - 5 )

= 6 C 5* ( 0.7 ) 5 * 0.3

= 6 * ( 0.7 ) 5 * 0.3

Similarly,

Probability of Success of Peter = 0.6

Probability of Failure of Peter = 1 - 0.6 = 0.4

For Peter to win 1st prize he has win all 6 times. So,

Probability that Peter wins the 1st prize = 6 C 6 * ( 0.6 ) 6 * ( 0.4 ) 6

= ( 0.6 ) 6 * ( 0.4 ) 6

Similarly,

For Peter to win 2nd prize he has win 5 times out of 6 chances. So,

Probability that Peter wins the 2nd prize = 6 C 5* ( 0.6 ) 5 * ( 0.4 ) ( 6 - 5 )

= 6 C 5* ( 0.6 ) 5 * 0.4

= 6 * ( 0.6 ) 5 * 0.4

Now,

Probability that John wins the 2nd prize and Peter wins the 1st prize

= Probability that John wins the 2nd prize * Probability that Peter wins the 1st prize

= 6 * ( 0.7 ) 5 * 0.3 * ( 0.6 ) 6 * ( 0.4 ) 6

And,

Probability that John wins the 1st prize and Peter wins the 2nd prize

= Probability that John wins the 1st prize * Probability that Peter wins the 2nd prize

= ( 0.7 ) 6 * ( 0.3 ) 6 * 6 * ( 0.6 ) 5 * 0.4

Probability that each of them has won a prize

= Probability that John wins the 2nd prize and Peter wins the 1st prize OR Probability that John wins the 2nd prize and Peter wins the 1st prize

= Probability that John wins the 2nd prize and Peter wins the 1st prize + Probability that John wins the 2nd prize and Peter wins the 1st prize

= { 6 * ( 0.7 ) 5 * 0.3 * ( 0.6 ) 6 * ( 0.4 ) 6 } + { ( 0.7 ) 6 * ( 0.3 ) 6 * 6 * ( 0.6 ) 5 * 0.4 }

So,

Probability that John wins the 2nd prize and Peter wins the 1st prize, given that each of them has won a prize

=   Probability that John wins the 2nd prize and Peter wins the 1st prize

Probability that each of them has won a prize

= 6 * ( 0.7 ) 5 * 0.3 * ( 0.6 ) 6 * ( 0.4 ) 6

  { 6 * ( 0.7 ) 5 * 0.3 * ( 0.6 ) 6 * ( 0.4 ) 6 } + { ( 0.7 ) 6 * ( 0.3 ) 6 * 6 * ( 0.6 ) 5 * 0.4 }

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