Question

according to masterfoods, the company that manufactures m&m’s, 12% of peanut m&m’s are brown, 15% are yellow, 12% are red, 23% are blue, 23% are orange and 15% are green. (round your answers to 4 decimal places where possible)
a. compute the probability that a randomly selected peanut m&m is not red.
b. compute the probability that a randomly selected peanut m&m is orange or red.
c. compute the probability that three randomly selected peanut m&m’s are all green.
d. if you randomly select six peanut m&m’s, compute that probability that none of them are red.
e. if you randomly select six peanut m&m’s, compute that probability that at least one of them is red.

Explanation:

Step1: Recall probability rules

The probability of an event not occurring is 1 minus the probability of the event occurring. For mutually - exclusive events, the probability of either event occurring is the sum of their probabilities. For independent events, the probability of multiple events all occurring is the product of their individual probabilities.

Step2: Calculate probability for part a

The probability that a peanut M&M is red is $P(R)=0.12$. The probability that it is not red is $P(\text{not }R)=1 - P(R)=1 - 0.12 = 0.8800$.

Step3: Calculate probability for part b

The probability that a peanut M&M is orange is $P(O)=0.23$ and the probability that it is red is $P(R)=0.12$. Since they are mutually - exclusive events, $P(O\cup R)=P(O)+P(R)=0.23 + 0.12=0.3500$.

Step4: Calculate probability for part c

The probability that a single peanut M&M is green is $P(G)=0.15$. Since the selections are independent, the probability that three are green is $P(G_1\cap G_2\cap G_3)=P(G)\times P(G)\times P(G)=0.15\times0.15\times0.15 = 0.0034$.

Step5: Calculate probability for part d

The probability that a single peanut M&M is not red is $P(\text{not }R)=0.88$. Since the selections are independent, the probability that none of six are red is $P(\text{none red})=(0.88)^6\approx0.4645$.

Step6: Calculate probability for part e

The probability that at least one is red is the complement of the event that none are red. So $P(\text{at least one red}) = 1 - P(\text{none red})=1-(0.88)^6\approx1 - 0.4645 = 0.5355$.

Answer:

a. 0.8800
b. 0.3500
c. 0.0034
d. 0.4645
e. 0.5355