Section10.3Bernoulli Trials

Suppose we have a jar with $7$ marbles, four of which are red and three are blue. A marble is drawn at random and we record whether it is red or blue. The probability $p$ of getting a red marble is $4/7\text{;}$ and the probability of getting a blue is $1-p=3/7\text{.}$

Now suppose the marble is put back in the jar, the marbles in the jar are stirred, and the experiment is repeated. Then the probability of getting a red marble on the second trial is again $4/7\text{,}$ and this pattern holds regardless of the number of times the experiment is repeated.

It is customary to call this situation a series of Bernoulli trials. More formally, we have an experiment with only two outcomes: success and failure. The probability of success is $p$ and the probability of failure is $1-p\text{.}$ Most importantly, when the experiment is repeated, then the probability of success on any individual test is exactly $p\text{.}$

We fix a positive integer $n$ and consider the case that the experiment is repeated $n$ times. The outcomes are then the binary strings of length $n$ from the two-letter alphabet $\{S,F\}\text{,}$ for success and failure, respectively. If $x$ is a string with $i$ successes and $n-i$ failures, then $P(x)=\binom{n}{i}p ^i(1-p)^{n-i}\text{.}$ Of course, in applications, success and failure may be replaced by: head/tails, up/down, good/bad, forwards/backwards, red/blue, etc.

Example10.12.

When a die is rolled, let's say that we have a success if the result is a two or a five. Then the probability $p$ of success is $2/6=1/3$ and the probability of failure is $2/3\text{.}$ If the die is rolled ten times in succession, then the probability that we get exactly four successes is $C(10,4)(1/3)^4 (2/3)^{6}\text{.}$

Example10.13.

A fair coin is tossed $100$ times and the outcome (heads or tails) is recorded. Then the probability of getting heads $40$ times and tails the other $60$ times is

\begin{equation*} \binom{100}{40}\left(\frac{1}{2}\right)^{40}\left(\frac{1}{2}\right)^{60} =\frac{\binom{100}{40}}{2^{100}}. \end{equation*}
Discussion10.14.

Bob says that if a fair coin is tossed $100$ times, it is fairly likely that you will get exactly $50$ heads and $50$ tails. Dave is not so certain this is right. Carlos fires up his computer and in few second, he reports that the probability of getting exactly $50$ heads when a fair coin is tossed $100$ times is

\begin{equation*} \frac{12611418068195524166851562157}{158456325028528675187087900672} \end{equation*}

which is $.079589\text{,}$ to six decimal places. In other words, not very likely at all. Xing is doing a modestly more complicated calculation, and he reports that you have a $99$% chance that the number of heads is at least $20$ and at most $80\text{.}$ Carlos adds that when $n$ is very large, then it is increasingly certain that the number of heads in $n$ tosses will be close to $n/2\text{.}$ Dave asks what do you mean by close, and what do you mean by very large?