What is the probability that Billy scored the other one?

Question

Billy and Lizzy have probabilities 0.3 and 0.2 respectively to score a goal in a given soccer

match. Two goals are scored, one of which was from Lizzy.

I was a bit confused cause from what I know, probabilities have to add up to 1, I did the probability tree and I'm not too sure if it's correct, if anyone knows would be much appreciated.

Answer 1

I agree that probabilities should add up to 1, as that represents the probability of all possible outcomes.

My interpretation of this question is that there are there are four distinct outcomes to consider:

1) Neither Billy nor Lizzy score

2) Billy scores, and Lizzy does not score

3) Lizzy scores, and Billy does not score

4) Both Billy and Lizzy score

P(Neither Billy nor Lizzy score) = P(Billy does not score)*P(Lizzy does not score) = 0.7*0.8 = 0.56

P(Billy scores, and Lizzy does not score) = P(Billy scores)*P(Lizzy does not score) = 0.3*0.8 = 0.24

P(Lizzy scores, and Billy does not score) = P(Billy does not score)*P(Lizzy scores) = 0.7*0.2 = 0.14

P(Both Billy and Lizzy score) = P(Billy scores)*P(Lizzy scores) = 0.3*0.2 = 0.06

Note that 0.56 + 0.24 + 0.14 + 0.06 = 1 (thus, this accounts for all outcomes).

So, if we are only considering Billy and Lizzy, and if we assume that their abilities to score are independent of each other, then the probability that both of them score in a single match is 0.06.

Hope this helps!