Physics and Maths Tutoring from Michael A Smith.

Plants can have a number of flowers. After looking at a large population we can determine the probability of each count of flowers, then use this to determine an estimate of the number in a small.

The Problem:

A species of plant can have one to five flowers. The probability of each number of flowers is given by:

Number of Flowers	1	2	3	4	5
Probability	0.12	0.24	0.23	0.20	?

Peter plants 96. How many of Peter's plants will have three, or more, flowers?

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Solution:

Firstly, our table is missing the probability for five flowers. This can be found by knowing that the probabilities of all the outcomes sum to one.

$0.12 + 0.24 + 0.23 + 0.20 + P (5 flowers) = 1$

$P (5 flowers) = 1 - 0.79$
∴ $P (5 flowers) = 0.21$

The question asks for the number of plants with three, or more, flowers, i.e., the number of planted times the probability of three or more flowers,

$N P (n \geq 3)$

where $N$ is the number planted and $n$ is the number of flowers.

$N = 96$ from the question;

and

$P (n \geq 3) = P (n = 3) + P (n = 4) + P (n = 5)$

$P (n \geq 3) = 0.64$

So, the number of plants with three, or more, flowers is $96 \times 0.64 = 61.44$

This is a rather unusual count for the number of plants, because this is the mathematical answer. We now have to think of this number in the context of the problem. Rounding to the nearest integer we get 61.

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