Math 409/409GHistory of Mathematics

Bernoulli Trials

A probability experiment is a Bernoulli trials experiment if:

• The experiment is performed a fixed number of times (called trials).

• The trials are independent. (Outcome of one trial has no affect on the outcome of other trials.)

• Each trial has only two mutually exclusive outcomes (success of failure). And

• The probability of a success is fixed for each trial in the experiment.

If n the number of trials in the experiment,

p the probability of a success, and

q the probability of a failure,

Then

Where is a binomial coefficient.

.Pr[ successes & failures] r n rnr n r p q

r

!

! !

n n

r r n r

Since the outcomes of a success and of a failure are mutually exclusive events (events can’t happen at the same time), the probability of a failure, q, is q 1 – p where p is the probability of a success. So

can be stated more concisely as:

!(1 ) .

!( )!Pr[exactly successes] r n rn

r p pr n r

.Pr[ successes & failures] r n rnr n r p q

r

Example

According to Nielsen Media Research, 75% of all US households have cable TV.

In a random sample of 15 households, what (approximately to 4 decimal places) is the probability that:

1. Exactly 13 have cable?

2. At least 13 have cable?

3. Fewer than 13 have cable?

75% of all US households have cable TV. In a random sample of 15 households, what is the probability that …

This experiment describes a Bernoulli trials experiment since:– The experiment is performed a fixed number of

times (n 15, the number of households).– The trials are independent. (One household

having or not having cable has no bearing on another household having or not having cable.)

– Each trial has only 2 mutually exclusive outcomes. (Does or does not have cable.) And

– The probability of having cable is fixed for each household. (p 0.75 ¾)

75% of all US households have cable TV. In a random sample of 15 households, what is the probability that …

So we can use Bernoulli’s formula

with n 15 and p ¾. This gives

!(1 )

!( )!Pr[exactly successes] r n rn

r p pr n r

1515! 3 1

.!(15 )! 4 4

Pr[exactly cable homes]r r

rr r

1. What (approx. to 4 decimal places) is the probability that exactly 13 households have cable?

In this case, since we want the exact number of cable homes, r 13 in the formula. So

1515! 3 1

!(15 )! 4 4Pr[exactly cable homes]

r r

rr r

13 115

1

3

3 2

15! 3 1

!(15 )! 4 4

15! 3 10.155907045... 0.1559

13!2! 4 4

13 13Pr[exactly cable h13 omes]

2. What (approx. to 4 dec. places) is the probability that at least 13 households have cable?

The phrase at least means greater than or equal to. So in the context of this problem, “at least 13” means “exactly 13 or exactly 14 or exactly 15.”

1515! 3 1

!(15 )! 4 4Pr[exactly cable homes]

r r

rr r

13 13 14 15

13 14 15

0.1559 0.0668 0.0134 0.2361

Pr[ ] Pr[ or or ]

Pr[ ] Pr[ ] Pr[ ]

at least r r r

r r r

3. What (approx. to 4 dec. places) is the probability that fewer than 13 households have cable?

The phrase fewer than is the same as the phrase less than, which in turn is equivalent to the phrase not greater than or equal to.

So in the context of this problem, the event that r < 13 is the compliment of the event that r 13.

13 1 13

1 0.2361 0.7639

Pr[ ] Pr[ ]fewer than at least

This ends the lesson on

Bernoulli Trials