Exercise 38 Page 745

If we have n repeated independent trials, each with a probability of success p and a probability of failure q, then the Binomial Probability of x successes in the n trials can be found by the following formula. P(x)= _nC_x p^x q^(n-x) We are told that p= 0.2. With the value of p and knowing that p+q=1, we can calculate the value of q.

p+q=1

Substitute

p= 0.2

0.2+q=1

SubEqn

LHS-0.2=RHS-0.2

q= 0.8

We are also told that x= 4 and n= 7. We can substitute all of these values into the formula and evaluate the right-hand side.

P(x)= _nC_x p^x q^(n-x)

SubstituteValues

Substitute values

P( 4)= _7C_4 ( 0.2)^4 ( 0.8)^(7- 4)

Next, let's recall the formula to calculate _nC_x. _nC_x =n!/x!(n-x)! Therefore, we can substitute 7!4!(7-4)! for _7C_4.

P(4)= _7C_4 (0.2)^4 (0.8)^(7-4)

Substitute

_7C_4= 7!/4!(7-4)!

P(4)= 7!/4!(7-4)! (0.2)^4 (0.8)^(7-4)

▼

Evaluate right-hand side

SubTerms

Subtract terms

P(4)= 7!/4!(3!) (0.2)^4 (0.8)^3

Write as a product

P(4)= 7(6)(5)(4!)/4!(3)(2)(1) (0.2)^4 (0.8)^3

CancelCommonFac

Cancel out common factors

P(4)= 7(6)(5)(4!)/4!(3)(2)(1) (0.2)^4 (0.8)^3

SimpQuot

Simplify quotient

P(4)= 7(6)(5)/3(2)(1) (0.2)^4 (0.8)^3

Multiply

Multiply

P(4)= 210/6 (0.2)^4 (0.8)^3

CalcQuot

Calculate quotient

P(4)=35 (0.2)^4 (0.8)^3

CalcPow

Calculate power

P(4)=35 (0.0016) (0.512)

Multiply

Multiply

P(4)=0.028672

RoundDec

Round to 5 decimal place(s)

P(4)≈ 0.02867

If the probability of success for each trial is 0.2, then the probability of 4 successes in 7 trials is about 0.02867.