Probability of Wayne & Mario Both Missing Penalty Shot

[Inertialforce]

Homework Statement

Two hockey players, Wayne and Mario, each independently take a penalty shot. Wayne has a 7/10 chance of scoring, while Mario has a 3/5 chance of scoring. What is the probability that;

a) they both miss

Homework Equations

P(A and B) = P(A) x P(B)

The Attempt at a Solution

For this question I found out what the complements (P(not mario) and P(not Wayne)) for each were and then I used the formula for independent events [P(A and B) = P(A) x P(B)] to find my answer and the answer that I got was correct. However, I was wondering why I cannot just go:

P(both miss) = 1- P(both score)

I tried this method and the answer that came up was different than the answer that I got from my first method of trying to solve this question (using the complements). Could someone explain why I cannot use "P(both miss) = 1- P(both score)" to solve this question?

 

[danago]

Homework Statement

Two hockey players, Wayne and Mario, each independently take a penalty shot. Wayne has a 7/10 chance of scoring, while Mario has a 3/5 chance of scoring. What is the probability that;

a) they both miss

Homework Equations

P(A and B) = P(A) x P(B)

The Attempt at a Solution

For this question I found out what the complements (P(not mario) and P(not Wayne)) for each were and then I used the formula for independent events [P(A and B) = P(A) x P(B)] to find my answer and the answer that I got was correct. However, I was wondering why I cannot just go:

P(both miss) = 1- P(both score)

I tried this method and the answer that came up was different than the answer that I got from my first method of trying to solve this question (using the complements). Could someone explain why I cannot use "P(both miss) = 1- P(both score)" to solve this question?

P(both score) is the probability that both mario AND wayne score. The complement of P(both) is the probability that not both of them score, so it could mean that one of them scores and the other doesnt, not necessarily that they both miss.

i.e.
$$1-P(A \cap B) = P(\overline{A\cap B}) \neq P(\overline{A} \cap \overline{B})$$

 

[HallsofIvy]

1 - P(A)(P(B) is "1- probability they both score" and so is probability the do not BOTH score. But "not both scoring" is not the same as "both do not score". "Not both scoring" includes one scores and the other does not.