Question

On the package for a certain brand of spinach seeds there is a guarantee that, if the printed instructions are followed, of planted seeds will germinate. A random sample of seeds is chosen. If these seeds are planted according to the instructions, find the probability that of them germinate.

Do not round your intermediate computations, and round your answer to three decimal places.

Answer 1

Explanation:

so, overall, 63% will grow.

that means, every seed has a 63% chance to grow. in other words, the probabilty to grow is 0.63 (as 63% means 63 out of 100 = 0.63).

and a chance of

1 - 0.63 = 0.37

to not grow.

fewer than 8 out of 9 seeds means not 8 and not 9.

it is

1 - P(8) - P(9)

with P(x) means the probability that exactly x seeds out of the 9 grow/germinate.

the chance for a group of exactly 8 out of 9 seeds to grow is then the probabilty for 8 to grow and 1 not to grow :

0.63⁸ × 0.37 = 0.009181764...

but these are 8 seeds out of 9.

and we calculated the probability for only one possible grouping.

how many groups of 8 can we build out of 9 ?

as the sequence does not matter, and we cannot have repetitions (every seed can only count as 1), we have combinations

C(9, 8) = 9! / (8! × (9-8)!) = 9!/8! = 9

so, the total probability P(8) for exactly 8 seeds out of 9 to grow/germinate is

9 × 0.63⁸ × 0.37 = 9 × 0.009181764... = 0.082635875... ≈

≈ 0.083

now, the chance for a group of exactly 9 out of 9 seeds to grow is then the probabilty P(9) for all 9 to grow :

0.63⁹ = 0.015633814... ≈ 0.016

the probabilty that fewer than 8 out of the 9 will grow/germinate is

1 - 0.082635875... - 0.015633814... = 0.901730311... ≈

≈ 0.902