Probability John Loses & Meghan Wins: 9/55

[zorro]

Homework Statement

The probability that John wins a game is 4/5 and,
independently, the probability that Meghan wins is 9/11.
What is the probability that Meghan wins and
John loses the game?

The Attempt at a Solution

I calculated it as 18/55 but the answer given is 9/55. Why is it so? Is the order of winning not important here? Why?

 

[ehild]

what is the probability that John loses the game? :)

ehild

 

[zorro]

1/5.

 

[Ray Vickson]

If the "win" probabilities are independent, are the "lose" probabilities independent? Are the events that John loses and Meghan wins independent? (Suggestion: don't _guess_. Work out the responses from first principles, using the definitions of independence, etc. This issue comes up over and over again in many places, so it would be as well to understand it.)

RGV

 

[ehild]

So how come that you got 18/55 for the probability that John loses the game and Meghan wins it? The events are independent, it can be any game, they are not winners and losers in the same game at the same time.

ehild

 

[zorro]

I have seen questions like ' A bag contains 4 Red and 7 Blue balls. Two balls are picked up from the bag at random without replacement. What is the probability that one ball is blue and the other is red? '

Here we compute the probability of selecting one red and one blue ball (14/55) and then multiply it by 2 because the first one can be red or blue. I applied the same logic in the question above. John can be the first or the second loser.

 

[ehild]

Picking out two balls from the same bag without replacement are not independent events. You can select a red one first with probability 4/11 and then a blue ones from the rest with probability 7/10 or a blue first with probability 7/11 and then a red one with probability 4/10, so the total probability is indeed 2*28/110.

The events in your problem are independent. That game is not played between Meghan and John alone as in this case the winner would determine the user so the probability that John wins and Meghan loses would be the same as the probability that John wins alone. That game can be something played by a lot of people many times. It can be, for example, writing a test successfully or failing it. The probability that John passes the tests is 4/5 and the probability that Meghan passes them is 9/11, and it is a new test now and one asks the probability that Meghan passes it and John fails. These events are totally independent, they are honest, do not look at each others tests...

ehild

 

[zorro]

I got it now. Thanks.