The Starfleet Academy soccer team excels on muddy fields in

The Starfleet Academy soccer team excels on muddy fields in the rain. The probability that the Starfleet Academy soccer team wins given that the field is muddy is 0.6. The probability that the Starfleet Academy soccer team wins given that the field is dry is 0.5. (a) If the forecast for rain on game day has an 20% chance of rain, what is the probability that the Starfleet Academy team wins? (Reminder: Define all notation. Only set up the calculation and replace any symbols with relevant numbers from the problem. You do not need to carry out the calculation.)

Solution

(a)

Let W shows the event that team win, M shows the event that field is Muddy and D shows the event the field is dry. So we have following probability:

P(W|M)=0.6

P(W|D)=0.5

Since there are 20% chance of rain so the probability that field will Muddy is

P(M)=0.20

and the probability that field will dry is

P(D)=1-0.20=0.80

So the probability that team will win is

P(W)=P(W|M)*P(M)+P(W|D)*P(D)=0.6*0.2+0.5*0.8=0.12+0.4=0.52

Hence, the required probability is 0.52.