Lesson on Binomials

The binomial distribution

A) An Introduction of binomial distribution

Drag and Drop

You can use ​

Placeholder

to work out ​

Placeholder

of events involving a number of separate stages.

A special case is where each stage has only ​

Placeholder

, and the probabilities of these two outcomes are the same at each stage (p and 1 - p).

Drag these tiles and drop them in the correct blank above

tree diagrams

probabilities

two outcomes of interest

Drag and Drop

Even when there are more than ​

Placeholder

, they can usually be grouped into ​

Placeholder

: those which satisfy the event of interest (often called ​

Placeholder

), and those which do not (​

Placeholder

).

At any stage, ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

two basic outcomes

two categories

'success'

'failure'

P(failure) = 1 — P(success)

Drag and Drop

For example, if the aim was to count how often a factor of 6 is thrown when

a die is rolled, then throws of any of 1, 2, 3 and 6 will result in 'success'.

So, in this case: P(success) = ​

Placeholder

and P(failure) = ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

Some patterns begin to emerge when these probabilities are shown on the branches of a tree diagram and the number of successes is counted.

The number of paths corresponds to the number of different ways you can order the three probabilities, in each case. The l, 3, 3, 1 sequence of the number of paths, for x = 0, 1, 2, 3, is one ofthe rows of Pascal's triangle, as shown on the right.

Match

Match the following

• $$1\times\left(\frac{2}{3}\right)^0\times\left(\frac{1}{3}\right)^3$$

• $$3\times\left(\frac{2}{3}\right)^1\times\left(\frac{1}{3}\right)^2$$

• $$3\times\left(\frac{2}{3}\right)^2\times\left(\frac{1}{3}\right)^1$$

• $$1\times\left(\frac{2}{3}\right)^3\times\left(\frac{1}{3}\right)^0$$

P(x = 0)

P(x = 1)

P(x = 2)

P(x = 3)

• P(x = 0)

• P(x = 1)

• P(x = 2)

• P(x = 3)

Drag and Drop

If the tree diagram is extended to six stages (that is, ​

Placeholder

), there would be ​

Placeholder

, and it would be cumbersome to work through the whole process like this.

Drag these tiles and drop them in the correct blank above

six throws of the die

64 different paths

However, the coefficients in Pascal's triangle and the patterns seen in the diagram with three stages can be used to write down expressions for the probabilities of getting 0, 1, 2, 3, 4, 5 or 6 successes.

Multiple Choice

When the number of stages (or trials) is large (> 6), it is hard to use Pascal's triangle to find the binomial coefficients.

The number of paths giving r occurrences out of n cases equals the number of ways of choosing r out of n, which is . . .

• $$C_r^n=\frac{n!}{\left(n-r\right)!r!}$$

• $$P_r^n=\frac{n!}{r!}$$

Drag and Drop

We need to have values for n, the number of trials, and for p, the probability of ​

Placeholder

on any one trial, in order for this to make sense, so there is a family of binomial distributions with two ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

'success'

parameters

Drag and Drop

The parameters of the binomial distribution are n (​

Placeholder

) and p (​

Placeholder

) on any one trial.

Drag these tiles and drop them in the correct blank above

the number of trials

the probability of a 'success'

Multiple Choice

If $$X\sim B\left(12,\ 0.2\right)$$ , find the probability that $$X=3$$ .

• $$P\left(X=3\right)=C_3^{12}\left(0.2\right)^9\left(0.8\right)^3$$

• $$P\left(X=3\right)=C_3^{12}\left(0.2\right)^3\left(0.8\right)^9$$

Dropdown

If 25 fair dice are rolled, find the probability that three 6s are seen.

If X = number of 6s seen, then ​

Select answer

Multiple Choice

If 25 fair dice are rolled, find the probability that three 6s are seen.

If X = number of 6s seen, then $$X\sim\left(25,\frac{1}{6}\right)$$ .

• $$P\left(X=3\right)=C_3^{25}\left(\frac{1}{6}\right)^3\left(\frac{5}{6}\right)^{22}$$  

• $$P\left(X=3\right)=C_3^{25}\left(\frac{1}{6}\right)^{22}\left(\frac{5}{6}\right)^3$$  

B) Mean and variance of the binomial distribution

Mean and variance of the binomial distribution

Drag and Drop

If $$X\sim B\left(10,\ 0.2\right)$$ , find the mean and variance of $$X$$ .

Given that n = ​

Placeholder

and p = ​

Placeholder

, so q =​

Placeholder

.

Therefore the mean is np = ​

Placeholder

and the variance is npq = ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

10

0.2

0.8

2

1.6

Dropdown

If $$X\sim B\left(80,\ 0.4\right)$$ , find the mean. variance, and the standard deviation of $$X$$ .

Therefore the mean is ​

Select answer

and the variance is ​

Select answer

, so the standard deviation is ​

Select answer

Dropdown

X is a binomial distribution with mean 8 and variance 6.4.

Then, the value of np = ​

Select answer

, npq = ​

Select answer

, q = ​

Select answer

, p = ​

Select answer

, and n = ​

Select answer

.