Probability in Game
You are off to soccer, and want to be the Goalkeeper, but that depends who is the Coach today:
with Coach Sam the probability of being Goalkeeper is 0.5 with Coach Alex the probability of being Goalkeeper is 0.3 Sam is Coach more often ... about 6 out of every 10 games (a probability of 0.6).
So, what is the probability you will be a Goalkeeper today?
The answer is 0.42.
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1 solution
Let's build a tree diagram. First we show the two possible coaches: Sam or Alex:
The probability of getting Sam is 0.6, so the probability of Alex must be 0.4 (together the probability is 1)
Now, if you get Sam, there is 0.5 probability of being Goalie (and 0.5 of not being Goalie):
If you get Alex, there is 0.3 probability of being Goalie (and 0.7 not):
The tree diagram is complete, now let's calculate the overall probabilities. Remember that:
P(A and B) = P(A) x P(B|A)
Here is how to do it for the "Sam, Yes" branch:
(When we take the 0.6 chance of Sam being coach and include the 0.5 chance that Sam will let you be Goalkeeper we end up with an 0.3 chance.)
But we are not done yet! We haven't included Alex as Coach:
An 0.4 chance of Alex as Coach, followed by an 0.3 chance gives 0.12
And the two "Yes" branches of the tree together make:
0.3 + 0.12 = 0.42 probability of being a Goalkeeper today