Binomial Distribution

Binomial distribution is the probability distribution for a binomial experimentHelps find probabilities when there are a large number of trials

Characteristics

Each trial has two outcomes or can be reduced to two outcomes.

Ex: when rolling a die, getting odd or even, or rolling a two or not a two

There is a fixed number of trials Outcomes of each trial must be

mutually exclusive P(s) – probability of success must

remain same for each trial (independent events)

Examples of Binomial Experiments

Any experiment with only two possible outcomes

Tossing a coin Type of child (boy or girl) Rolling a die (3 or not 3) Red or Black card

A coin is tossed 3 times. Find the probability of getting exactly 2 heads.

Sample Space

H H H

H H T

H T H

H T T

T H H

T H T

T T H

T T T

Binomial Distribution

X = # of heads 2 Not 2

P(x) 3/8 5/8

Mean and Standard Deviation of a Binomial Distribution

Formula for mean:

Formula for standard deviation:

Reminder: n is the # of trials (or sample size) p is the probability of success q is 1 - p

np

npq

Example 1: mean and standard deviation

A card is selected from a standard deck of cards and then replaced. Find the mean and standard deviation of the # of aces selected if you pick 20 cards.

n = 20 p = 4/52 q = 48/52

420 1.5

52

4 4820 1.19

52 52

Example 2: The probability that a divorcee will

remarry within 3 years is 40%. In a sample of 200 divorcees, find the mean and standard deviation of the number of people that will remarry within 3 years.

n = 200 p = .40 q = .60

200 .40 80

200 .40 .60 6.93

Finding the Probability of a Binomial Distribution – Example 1

A coin is tossed 3 times. Find the probability of getting exactly 2 heads.Sample Space

H H H

H H T

H T H

H T T

T H H

T H T

T T H

T T T

Binomial Distribution

X = # of heads 2 Not 2

P(x) 3/8 5/8

Consider: What if we tossed the coin 20 times? 220 = sample space = 1,048,576 possibilities

Formula

Formula:

x = number of successes (what you are looking for)

n = number of trials, p = probability of success on a single

trial q = probability of failure on a single

trial (q = 1 – p)

( ) x n xn xP x C p q

Using the formula: Example 1

A coin is tossed 3 times. Find the probability of getting exactly two heads.

x = 2 heads n = 3 p = ½ q = ½

2 13 2(2 ) *(1/ 2) *(1/ 2)P heads C

(2 ) 3 / 8P heads

Example 2

A coin is tossed 20 times. Find the probability of getting exactly two heads.

x = 2 heads n = 20 p = ½ q = ½ 2 18

20 2(2 ) *(1/ 2) *(1/ 2)P heads C190

1048576

Example 3: In a survey, 75% students said the

courts show too much concern for criminals. Find the probability that 3 out of 7 randomly selected students will agree with the statement.

n = 7 x = 3 p = .75 q = .25

3 4

7 3(3) * .75 * .25P C

.06

Modification to Example 3

Find the probability that at most 2 out of 7 will agree.

0 7

7 0(0) * .75 * .25 .00006P C 1 6

7 1(1) * .75 * .25 .001P C

2 5

7 2(2) * .75 * .25 .012P C

.012 .001 .00006 .01306

x = 0, 1, 2

Example 4:

A coin is tossed 8 times. Find the probability of getting at least 2 heads.

n = 8x = 2, 3, 4, 5, 6, 7, 8

p = 1/2 q = 1/2

Easier to use the complement rule: Find probability of 0 or 1 head and then subtract from 1. x = 0, 1

Example 4: Answer

0 8

8 0

1 1* * .004

2 2C

1 7

8 1

1 1* * .03125

2 2C

1 .03525 .96475

Example 5:

If the probability is .40 that a divorcee will remarry within 3 years, find the probabilities that of 10 divorcees:

(a) at least 8 will remarry within 3 years

(b) at least 2 will remarry within 3 years (a)P(at least 8) = .0131

(b)P(at least 2) = .954

Normal Approximation to the Binomial Distribution

A coin is tossed 50 times. Find the probability of getting at least 20 heads.

x = 20 or more heads n = 50 p = ½ q = ½

We can use the normal distribution to find probabilities like this as long as the following conditions apply:

It is a binomial distribution (properties) Probability of a success must be close to

.5 If both np and nq are greater than or equal

to 5

From Coin Toss Example

A coin is tossed 50 times. Find the probability of getting at least 20 heads.

x = 20 or more heads n = 50 p = ½ q = ½

np = 50(.5) = 25nq = 50(.5) = 25 both greater than .5

Calculate and

Find continuity correction factor ( .5) summary on page 313 in text

and np npq

Find z-score(s) – Use regular z-score formula

Determine area to be shaded and find it!

Example 2

Of the members of a bowling league, 10% are widowed. If 200 bowling league members are selected, find the probability that more than 10 will be widowed.

Example 3:

If a baseball player’s batting average is .320 find the probability that the player will get

a) at most 26 hits in 100 times at bat.

b) exactly 26 hits in 100 times at bat.