# probability of the next number of events occurring when the true probability is known [duplicate]

Possible Duplicate:
Probability of getting between

What is the probability that at least 24 of the next 50 people like to swim when the true probability of people liking to swim is 35%?

I thought it might be a conditional probability but I'm not really sure...

## marked as duplicate by whuber♦Dec 13 '11 at 6:07

• Under various conditions, this could be modeled as a tail probability for a binomial distribution: The number of people (out of $50$) who like to swim is a binomial random variable $X$ with parameters $(50,0.35)$ and you are asked for $P\{X \geq 24\}$. The Demoivre-Laplace approximation (or Central Limit Theorem) could be applied. – Dilip Sarwate Dec 12 '11 at 20:11
• Not sure how I would work it with that new piece of information. – J M Dec 12 '11 at 20:17
• Wait a while and most likely someone will tell you the exact formula in R to calculate the desired probability. – Dilip Sarwate Dec 12 '11 at 21:22

Let $x_i$ be the decision of person $i$, define $x_i=1$ if she wants to swim and define $P(x_i=1)=0.35=p_i$.

Then you want to find the probability that $Y=\sum_{i=1}^{50} x_i \geq 24$.

and

$$P(Y \geq 24) = \sum_{i=24}^{50}C{{n}\choose{i}}p_i^i(1-p_i)^{n-i}$$

that is, all the possible combinations giving more than 24 people willing to swim.

NB: this assumes, of course, that the decision of each person is iid.

This problem is intended to be answered as a Binomial as Dilip answered. In many software, the cumulative distribution is given so P(X >= 24) = 1 - P(X <24) but here is where you need to be careful P(X < 24) = P(X <=23). If normal approximation is used, then the estimate can be made more precise with continuity correction. In this case, use the normal distribution to compute P(X >= 23.5) instead of P(X >= 24). Finally, these kind of problems are written silly in that the "next" 50 people is presumed to be random and that the probability is 0.35 from person to person. That isn't likely when you take the "next" 50 people. The next 50 people could be the swim team!