Probability question help please?!!!!

The probability that a certain football club has all their first team players fit is 70%. When the club has a fully fit team it wins 90% of its home pages. When the first team is not fully fit it wins 40% of its home pages.
a) Calculate the probability that it will win its next home game.
b) Given that it did not win its last home game, find the probability that the team was fully fit.


thanks in advance! :)

Fill out the probability tree. (hard to do here but...)

Fitness--W/L

70% 90/10

30% 40/60

.7*.9=.63
.7*.10=.07
.3*.4=.12
.3*.6=.18
sum .63+.07+.12+.18=1 (this just checks that all the possibilities sum to 1)

The team wins if: its health and wins .7*.9=.63 or if its not health and still wins .3*.4=.12.
Sum these events for .75. The probability is 75% chance of winning.

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For a conditional probabiltiy, sum the way the event can happen divided by just the conditional probabilities.
The conditional is that the team did not win which can happen if: the team is healthy.7*.1 or the team is not fully health .3*.6. The total is .25 (note it is 1-the probability of winning from the first part)

so within this 25% probability, the team is fit with probability .7*.1=.07
divide the event by the condiitonal
.07/.25=.28

the probabilty is .28 that the team was healthy if it did not win its last game.