A stationery store has a box of 30 large fasteners and 20 small fasteners. They randomly...

A stationery store has a box of 30 large fasteners and 20 small fasteners. They randomly choose 20 of fasteners.

1. What is the probability that their sample contains exactly 10 large fasteners? Round your probability to 4-decimal places.

2. What is the probability that their sample contains exactly 10 large and 10 small fasteners or 11 large and 9 small fasteners? Round your probability to 4-decimal places.

Solutions

Expert Solution

A stationery store has a box of 30 large fasteners and 20 small fasteners.

They randomly choose 20 of fasteners.

1. What is the probability that their sample contains exactly 10 large fasteners?

P[ exactly 10 large fasteners ] = P[ exactly 10 large fasteners and exactly 10 small fasteners ] = Selecting 10 large fasteners out of 30*10 small fasteners out of 20 / Selecting any 20 fasteners out of 50

Selecting 10 large fasteners out of 30 = 30C10 = 30!/(20!*10!)

10 small fasteners out of 20 = 20C10 = 20!/(10!*10!)

Selecting any 20 fasteners out of 50 = 50!/(20!*30!)

P[ exactly 10 large fasteners ] =

P[ exactly 10 large fasteners ] = 0.117

2. What is the probability that their sample contains exactly 10 large and 10 small fasteners or 11 large and 9 small fasteners?

P[ exactly 10 large fasteners and exactly 10 small fasteners or 11 large and 9 small fasteners ] = P[ exactly 10 large fasteners and exactly 10 small fasteners ] + P[ 11 large and 9 small fasteners ]

P[ 11 large and 9 small fasteners ] = P[ exactly 11 large fasteners and exactly 9 small fasteners ] = Selecting 11 large fasteners out of 30*9 small fasteners out of 20 / Selecting any 20 fasteners out of 50

Selecting 11 large fasteners out of 30 = 30C11 = 30!/(19!*11!)

10 small fasteners out of 20 = 20C9 = 20!/(9!*11!)

Selecting any 20 fasteners out of 50 = 50!/(20!*30!)

P[ 11 large and 9 small fasteners ] =

P[ 11 large and 9 small fasteners ] =0.194

P[ exactly 10 large fasteners and exactly 10 small fasteners or 11 large and 9 small fasteners ] = P[ 11 large and 9 small fasteners ] + P[ exactly 10 large fasteners ]

P[ exactly 10 large fasteners and exactly 10 small fasteners or 11 large and 9 small fasteners ] = 0.117+0.194

P[ exactly 10 large fasteners and exactly 10 small fasteners or 11 large and 9 small fasteners ] = 0.311

orchestra answered 1 year ago

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