Question

Bags of plain M&Ms contain 24% blue candies. You have four bags of candy and select one candy from each bag. The random variable x represents the number of blue candies selected. The selected candy can be classified as either being blue (B) or not being blue (N). Answer the following questions based on this data. As always, you must show all work and formulas used in order to receive full credit. Round all decimals to three places 1. Draw a tree diagram for the above experiment, indicate the sample space, and use the Fundamental Counting Principle to specify the amount of outcomes in the sample space. As a hint, your sample space and tree diagram should illustrate the choices of an M&M either being ??? or not blue and should reflect the possible orders of this happening for each bag selection. (12 points) 2. Is this a binomial experiment? If so, explain why and identify the numerical values for the variables required. (8 points) 3 Construct a discrete probability distribution for this experiment. In order to find the probability for each value, you need to use the appropriate discrete probability formula separately for each x-value; therefore, you should be using the formula five different times. For full credit, along with showing your table for the discrete probability distribution, you must separately show the setup of the probability formula and your calculations for each x-value. (16 points 4. For this experiment, find a The mean amount of blue candies that will be selected in this experiment (6 points) b. The standard deviation for the amount of blue candies that will be selected in this experiment. ( 5. Using the probability distribution from question 3 that you've already constructed, find the following probabilities. You do not need to show the probability formula again since you should have already s your work for that in question 3. However, you should be showing the notation of what is being asked hown along with the values you are using from your probability distribution table and what you are doing with them to get your final answer Find the probability that at least three candies will be blue in this experiment 6points) Find the probability that fewer than two candies will be blue in this experiment. (6 points) a b.

Answer 1

Sol:

1:

Since each bag is independent from other so probability of getting blue ball time is

P(B) = 0.24, P(N) = 1 -0.24 = 0.76

Each ball can be blue or not blue so according to fundamental principe of counting total number of outcomes is

2 *2*2*2 = 16

Following is the tree diagram of the four draws (one from each bag):

Here outcome NNNB shows the balls from first three bags are not blue and ball from last bag is blue.

2)

Yes it is binomial experiment. Let X is a radnom variable shows the number of blue balls out of 4. Here X has binomial distribution with parameters n=4 and p=0.24.

The pdf of X is

$P(X=x)=\binom{4}{x}(0.24)^{x}(1-0.24)^{4-x},x=0,1,2,3,4$

3)

The probabilites are:

$P(X=0)=\binom{4}{0}(0.24)^{0}(1-0.24)^{4-0}=0.3336$

$P(X=1)=\binom{4}{1}(0.24)^{1}(1-0.24)^{4-1}=0.4214$

$P(X=2)=\binom{4}{2}(0.24)^{2}(1-0.24)^{4-2}=0.1996$

$P(X=3)=\binom{4}{3}(0.24)^{3}(1-0.24)^{4-3}=0.042$

$P(X=4)=\binom{4}{4}(0.24)^{4}(1-0.24)^{4-4}=0.0033$

Following table shows the required probability;

X	P(X=x)
0	0.3336
1	0.4214
2	0.1996
3	0.042
4	0.0033

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4:

The mean is

$\mu=np=4\cdot 0.24=0.96$

The standard deviation is

$\sigma=\sqrt{np(1-p)}=\sqrt{4\cdot 0.24\cdot 0.76}=0.8542$

5:

(a)

The probabiluty that at least three candies will be blue in this experiment is

$P(X\geq 3)=0.0453$

(b)

P(X < 2) = 0.755