4

I am the least mathematical person around, so apologies if this question is really dumb, but I'm trying to improve! I've been reading loads of examples everywhere but I'm having a hard time applying the logic/rules of probability to new problems.

Let's say you have to win three out of three rounds of a game in order to win a prize. It is a single player game.

The probability of a boy winning a round is $.25$, and the probability of a girl winning a round is .4. Winning one round doesn't influence the result of the next round. So if I haven't misunderstood, the probability of a girl winning a prize is $.4 \cdot .4 \cdot .4 = 0.064$ and the probability of a boy winning a prize is $.25 \cdot .25 \cdot .25 = 0.016$

Now, this is where I'm stuck. What's the overall probability of a person winning a prize if $50\%$ of the players are girls and $50\%$ of the players are boys? Is it just $.016 + 0.064 = 0.08$? Or should I be dividing by $2$ here somewhere given that it's $50$ percent boys and $50$ percent girls.

Thanks in advance for your help.

Daniele Tampieri
  • 9,727
Lalaca
  • 43
  • What do you mean by 'it's a 50/50 split for boys and girls'? – Joe Mar 07 '21 at 14:19
  • The rules are not clear. How do you win a prize? Also...$.25+.4=.65<1$. If neither boys nor girls win $35%$ of the time, what happens in those rounds? – lulu Mar 07 '21 at 14:21
  • Sorry, I've edited for clarity! It's a single player game and you have to win three rounds in a row to win a prize. 50% of the players are boys, 50% of the players are girls. I'm trying to work out what's the overall probability of a player winning a prize (i.e winning three rounds in a row).

    @lulu For the 35% of the time, no one wins a prize.

     – Lalaca Mar 07 '21 at 14:32
  • Ok. How many people are playing? When you say the probability that a boy wins a round is $.25$ do you mean "the probability that there is a winner who happens to be a boy is $.25$" or something else? At first reading, I thought you meant that the probability that a particular boy wins is $.25$ but I doubt that I was correct. – lulu Mar 07 '21 at 14:40
  • Oh, in the edit you say it is a one player game? But then how is the player selected? I suggest taking your time to think through what you really mean to ask and then editing your post for clarity. As it stands, it's not at all clear what the rules are. – lulu Mar 07 '21 at 14:45

2 Answers2

4

Hint: Would it be reasonable for the probability of a random person winning be higher than both the 'girl' probability and the 'boy' probability?

paw88789
  • 40,402
4

Let's say that instead, the probability of a boy winning a prize is $0.9$ and the probability of a girl winning a prize is $0.8.$

Then by your logic, the probability of a random person winning a prize is $0.9^3 \times 0.8^3 = 0.729 + 0.512 = 1.241,$ which is impossible because we can't have probabilities of more than $1$.

So you can't add probabilities like that.

What you can and should do is start by drawing a probability tree diagram for what you want. It should look like this:enter image description here

The probability of a randomly selected person winning the game is then: $0.5 \times 0.016 + 0.5 \times 0.064 = 0.04,\ $ or $\frac{1}{25}.$

Adam Rubinson
  • 20,052
  • Thank you so much, that is super clear! – Lalaca Mar 07 '21 at 15:23
  • You're welcome. If you think it answered your question, please feel free to upvote and accept it as an answer by selecting the green tick by the side of this answer. – Adam Rubinson Mar 07 '21 at 15:24
  • I can't upvote as I don't have enough reputation yet, but I've accepted it as answered. Thanks again! – Lalaca Mar 07 '21 at 15:26
  • Oh ok well thanks. – Adam Rubinson Mar 07 '21 at 15:30
  • Could I ask a follow up? If let's say I wanted to get the probability of a person who won a prize being a girl, would I be correct in applying bayes theorem like this: P(Winning|Girl) * P(girl) / P(Winning) (0.064 * 0.5) / 0.04 = 0.8. Meaning that if a prize is won there's an 80% chance it was a girl? – Lalaca Mar 07 '21 at 17:14
  • I'm no expert on Bayes' theorem, but yes I believe so. I certainly works in this case. – Adam Rubinson Mar 07 '21 at 17:23