Statistics Auestion Binomial Distribution

Question

A stationary store has decided to accept a large shipment of ball-point pens if an inspection of 17 randomly selected pens yields no more than two defective pens.

(a) Find the probability that this shipment is accepted if 10% of the total shipment is defective. (Use 3 decimal places.)

 

(b) Find the probability that this shipment is not accepted if 20% of the total shipment is defective. (Use 3 decimal places.)

Answer 1

You would want to use the formula for Bernoulli trial: (n,k)*P^k *(1-P)^(n-k) where:

P=probability of success

n=# of tirals

k= # of a specific outcome

and (n,k), also can be written an nCk, is the binomial coefficient and is equal to:

nCk= n!/[k!*(n-k)!]

For this problem, it will be easier to find the inverse or the chance of accepting the shipment ( meaning either 0, 1, or 2 defecting pens) and subtracting that likelyhood from 1.

A) P=.9 chance of working pen

1-P=.1=chance of defective pen

The case with 0 failures is (17,0)*.9¹⁷*.1⁰=[17!/(0!*(17-0)!]*.9¹⁷*1=(17!/17!)*.9¹⁷=1*.9¹⁷=.167

Repeating for 1 failure gives (17,1)*.9¹⁶*.1¹=17*.9¹⁶*.1¹=.315

Repeating for 2 failures gives (17,2)*.9¹⁵*.1²=136*.9¹⁵*.1²=.280

The probability of accepting a package of 17 pens .762

The probability of rejecting a package of 17 pens is 1-.762=.238

B) Repeat part A, but with P=.8 and (1-P)=.2

The result is .299 probability of accepting and a .701 chance of rejecting