Question

Before a long road trip alone, I put 13 chocolate chunk cookies and 9 snickerdoodles in a bag for snacks. After one hour of driving, you reach in the bag without looking and take out a cookie at random and eat it. After another hour, you take out another cookie (again randomly, without looking). Show the work!

1. Show the probability tree summarizing the probabilities of possible outcome of taking cookies out of the bag (this is two trial without replacement).

2. Use the tree to construct a discrete probability distribution for the number of chocolate chunk cookies I take out of the bag.

3. From the discrete probability distribution you constructed in #2, calculate the mean and standard deviation of the number of chocolate chunk cookies I take out of the bag (remember, because we’re looking at probabilities, mean and standard deviation have to be calculated differently than in the context of a numeric variable).

Answer 1

I have drawn probability tree diagram in the above image.

Total cookies = 13+9 = 22

chocolate chunk cookies(CC) = 13

snickerdoodles (SD) = 9

After first trial cookies will reduce by 1 so total cookies will be 21 and after the second trial it will be reduced by 2 so total will be 20 cookies.

Depending on which cookie has been selected at each stage the probability of picking that cooking is written in blue colour over the line. And in the square bracket are the number of cookies we started with.

2) Discrete probability distribution

The green number tells us the number of CC cookies selected at each path.

X	P(X)	P(X)
0	9/22*8/21*7/20	0.05454545
1	(9/22*8/21*13/20) + (9/22*13/21*8/20)+(13/22*9/21*8/20)	0.3038961
2	(13/22*12/21*9/20)+(13/22*9/21*12/20)+(9/22*13/21*12*20)	0.45584416
3	13/22*12/21*11/20	0.18571429

3. Expected value =

Standard deviation =

I have calculated above in following table:

X	P(X)	X*P(X)	(X-E(X))^2*P(X)
0	0.05454545	0	0.171271855
1	0.3038961	0.3038961	0.181117216
2	0.45584416	0.91168831	0.023696603
3	0.18571429	0.55714286	0.280054171
	SUM	1.77272727	0.376085673

E(X) = 1.7727

Variance = 0.3760

Standard deviation = sqrt(variance) = 0.6132