Question

source: www.pollingreport.com

1. m&m color distribution according to the manufacturer, m&m’s are produced and distributed in the following proportions: 13% brown, 13% red, 14% yellow, 16% green, 20% orange, and 24% blue. in a random sample of 12 m&m’s what is the probability of having 2 of each color?

Explanation:

Step1: Identify multinomial probability formula

The multinomial probability formula for this scenario is:
$$P = \frac{n!}{n_1!n_2!n_3!n_4!n_5!n_6!} \times p_1^{n_1}p_2^{n_2}p_3^{n_3}p_4^{n_4}p_5^{n_5}p_6^{n_6}$$
Where $n=12$, $n_1=n_2=n_3=n_4=n_5=n_6=2$, and $p_1=0.13, p_2=0.13, p_3=0.14, p_4=0.16, p_5=0.20, p_6=0.24$ are the probabilities of each color.

Step2: Calculate factorial term

Compute the factorial ratio:
$$\frac{12!}{(2!)^6} = \frac{479001600}{(2)^6} = \frac{479001600}{64} = 7484400$$

Step3: Calculate probability product

Compute the product of each probability raised to the 2nd power:
$$(0.13)^2(0.13)^2(0.14)^2(0.16)^2(0.20)^2(0.24)^2$$
$$=(0.0169)(0.0169)(0.0196)(0.0256)(0.04)(0.0576)$$
$$=0.0169^2 \times 0.0196 \times 0.0256 \times 0.04 \times 0.0576$$
$$=0.00028561 \times 0.0196 \times 0.0256 \times 0.04 \times 0.0576$$
$$=0.0000056 \times 0.0256 \times 0.04 \times 0.0576$$
$$=0.00000014336 \times 0.04 \times 0.0576$$
$$=0.0000000057344 \times 0.0576 \approx 3.303\times10^{-10}$$

Step4: Multiply the two results

Multiply the factorial ratio by the probability product:
$$7484400 \times 3.303\times10^{-10} \approx 0.00247$$

Answer:

$\approx 0.0025$ (or 0.25% when expressed as a percentage)