Question

A very large gardening business grows rose bushes for sale to garden stores around the world. The most popular colors are red, pink, and white. The business decides on 50% red roses, 30% pink, and 20% white. A gardener orders 10 rose bushes selected randomly from a huge field. Her primary interest is in pink roses. A good model for the number of bushes with pink roses is given by the binomial distribution.The probability of getting at least 3 pink rose bushes is: (a) 0.5. (b) 0.617. (c) 0.348. (d) 0.259.

Answer 1

Answer:

(b)0.617.

Step-by-step explanation:

1) Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

2) Solution to the problem

Let X the random variable of interest, on this case we now that:

$X \sim Binom(n=10, p=0.3)$

The probability mass function for the Binomial distribution is given as:

$P(X)=(nCx)(p)^x (1-p)^{n-x}$

Where (nCx) means combinatory and it's given by this formula:

$nCx=\frac{n!}{(n-x)! x!}$

And we want to find this probability:

$P(X \geq 3)=1-P(X$

$=1-[P(X=0)+P(X=1)+P(X=2)]$

And now we can find the individual probabilities on this way:

$P(X=0)=(10C0)(0.3)^0 (1-0.3)^{10-0}=0.02825$

$P(X=1)=(10C1)(0.3)^1 (1-0.3)^{10-1}=0.12106$

$P(X=2)=(10C2)(0.3)^2 (1-0.3)^{10-2}=0.23347$

$P(X \geq 3)=1-[0.02825+0.12106+0.23347]=0.61722$

And rounded the correct answer would be (b)0.617.