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# Binomial Conditions

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Age 16 to 18

Challenge Level

The binomial distribution is appropriate when we have the following setup:

We perform a fixed number of trials, each of which results in "success" or "failure" (where the meaning of "success" and "failure" is context-dependent). We also require the following two conditions:

If we let $X$ be the number of successful trials, then $X$ has a binomial distribution.

*When we have a situation which looks like it might be binomial, we need to check that all of these conditions hold before we can use the binomial distribution formulae!*

We're going to design some scenarios which have a fixed number of trials, each of which results in "success" or "failure", and $X$ is the number of successful trials.

Is the probability distribution of $X$ that of a binomial distribution in either of your scenarios? (That is, does it have the form $\mathrm{P}(X=r)=\binom{n}{r}p^r(1-p)^{n-r}$ for $r=0$, ..., $n$ for some choice of $n$ and $p$?)

*You might find it helpful to work on Binomial or Not? first.*

*This resource is part of the collection Statistics - Maths of Real Life*

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We perform a fixed number of trials, each of which results in "success" or "failure" (where the meaning of "success" and "failure" is context-dependent). We also require the following two conditions:

(i) each trial has an *equal probability of success*, and

(ii)*the trials are independent*.

(ii)

If we let $X$ be the number of successful trials, then $X$ has a binomial distribution.

We're going to design some scenarios which have a fixed number of trials, each of which results in "success" or "failure", and $X$ is the number of successful trials.

(a) Is it possible that only (ii) holds, but not (i)? That is, can you design a scenario where the probability of success is not equal for all the trials, even though they are independent?

(b) Is it possible that only (i) holds, but not (ii)? That is, can you design a scenario where the trials are not independent, even though each trial has equal probability of success?

(b) Is it possible that only (i) holds, but not (ii)? That is, can you design a scenario where the trials are not independent, even though each trial has equal probability of success?

Is the probability distribution of $X$ that of a binomial distribution in either of your scenarios? (That is, does it have the form $\mathrm{P}(X=r)=\binom{n}{r}p^r(1-p)^{n-r}$ for $r=0$, ..., $n$ for some choice of $n$ and $p$?)

The icon for this problem is by Matemateca (IME/USP)/Rodrigo Tetsuo Argenton, originally downloaded from Wikimedia Commons (and then adapted for NRICH), licensed under CC BY-SA 4.0.