Probability Question Rationale Threw Me - Double Probability of Winning a Single Game

I was given the question:

James plays $$2$$ games of tennis against his friend Shaun. For each game the probability of James winning is $$0.3$$

The games are independent. James thinks he is more likely to win at least one game than not. Is he right?

The rationale explained:

This is what you get when you double the probability of James winning a single game:

My question in trying to understand their rationale was: Are you actually doubling the probability of him winning a single game by virtue of the fact that the statement The games are independent exists?

The worked solution then goes on to say The probability of winning one game or more is the probability of winning both games plus the probability of winning the first OR second game.

i.e.

$$0.3×0.3 = 0.09+2(0.3 ×0.7) = 0.42$$

$$= 0.51$$

Therefore it is greater than half.

• Don't write things like $.3\times .3=.09+2(.3\times .7)=.42$....it's very confusing. – lulu Jan 22 at 12:31
• That said, your calculation looks sound. Another way to see it is that the probability that James loses both games is $.7^2=.49$, so the probability he wins at least one is $1-.49=.51$ As you remark, the quote you provide is too terse and obscure to derive much meaning from. – lulu Jan 22 at 12:33