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pete and sean decide to raise money for a charity by having a carnival in their backyard. in one of the games that they set up, the probability that a person will win is 0.4. if robyn plays that game nine times, what is the probability that she wins exactly four times? A)9C5^5 (0.4)^5 (0.4)^5 B)9C4 (0.5)^4 (0.5)^5 C) 9C4 (0.4)^4 (0.6)^5 D)9C5 (0.4)^5 (0.6)^4 and how did you get really confused on probability formula thx

Probability
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what does 9C5^5 mean?
oh it's suppose to be 9C5 (0.4)^5 (0.4)^4 sorry
the answer anyway is \[(0,4)^4 * (0,6)^5 * \frac{ 9! }{ 4!5! }\] where (0.4) is the probability to win and (0.6) is 1 - 0.4 equals the probability to lose. Do you need more explanations why this is the answer?

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I guess 9C4 means \[\left(\begin{matrix}9 \\ 4\end{matrix}\right) = \frac{ 9! }{ 4!5! }\] then answer C is equivalent to my solution
why do you substract it by 1
well, what does the probability of 1 mean? It means that the case you're looking at is going to happen EVERY time (100% of the tries). Let p be the probability to win, and q be the probability to lose. In this game we assume that anytime you play, you either win or lose. This OR translates into the language of probability as a '+', that means p+q = 1. Because when you look at all possible cases and add them , that is equivalent to calculating the probability of one of the possible cases happening - which happens 100% of the time, hence the probability is 1. In this example, you the probability to win (p) and the probability to lose (q) are all the possible cases, hence p + q = 1 ==> q = 1-p
okay what do the exponents represents
They're kind of the translation of 'AND' into the language of probability. For example, if you tried once to win the game you had a probability of 0.4. If you tried twice, the probability to win twice would be (0.4) * (0.4) = (0.4)^2. The probability to play 9 times and win 9 games would be (0.4)^9 Similarly, if you played 5 times and lost all 5 games the probability for that would be (0.6)^5 Can you see the scheme here?
yeah! okay that pretty much sums most of my confusion thank you so much!

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