In a bag there are 8 red jelly beans and 7 green jelly beans. The 4 jelly beans are randomly chosen from the bag without replacement.
a)create a probability distribution table for choosing red jelly beans
b)what is the probability that at least 2 chosen jelly beans are green given that they are not all red?
Thanks.
a) To create a probability distribution table for choosing red jelly beans, we can list the possible outcomes and their corresponding probabilities.
Let's denote R for red jelly beans and G for green jelly beans.
Possible Outcomes:
1. Choose 0 red jelly beans: GGGG
2. Choose 1 red jelly bean: RGGG, GRGG, GGRG, GGGR
3. Choose 2 red jelly beans: RRGG, RGRG, RGGR, GRRG, GRGR, GGRG
4. Choose 3 red jelly beans: RRGG, RGRG, RGGR, GRRG, GRGR, GGRG
5. Choose 4 red jelly beans: RRRG, RRGR, RGRG, RGGR, GRRR
Probabilities:
1. P(R=0) = (7/15) * (6/14) * (5/13) * (4/12) = 0.0969
2. P(R=1) = (8/15) * (7/14) * (6/13) * (5/12) = 0.1633
3. P(R=2) = (8/15) * (7/14) * (6/13) * (5/12) = 0.1633
4. P(R=3) = (8/15) * (7/14) * (6/13) * (5/12) = 0.1633
5. P(R=4) = (8/15) * (7/14) * (6/13) * (5/12) = 0.1633
This gives us the probability distribution table for choosing red jelly beans:
R | P(R)
-----------------
0 | 0.0969
1 | 0.1633
2 | 0.1633
3 | 0.1633
4 | 0.1633
b) To find the probability that at least 2 chosen jelly beans are green, given that they are not all red, there are two cases to consider:
Case 1: 2 green jelly beans are chosen.
P(2 green jelly beans) = (7/15) * (6/14) * (8/13) * (7/12) = 0.0800
Case 2: 3 green jelly beans are chosen.
P(3 green jelly beans) = (7/15) * (6/14) * (5/13) * (8/12) = 0.0533
To find the overall probability, we sum the probabilities of the two cases:
P(at least 2 green) = P(2 green jelly beans) + P(3 green jelly beans) = 0.0800 + 0.0533 = 0.1333
Therefore, the probability that at least 2 chosen jelly beans are green, given that they are not all red, is 0.1333.
a) To create a probability distribution table for choosing red jelly beans, we need to find the probabilities of choosing 0, 1, 2, 3, or 4 red jelly beans.
Let's calculate each probability step-by-step:
Step 1: Calculate the total number of ways of choosing 4 jelly beans out of 15 (8 red + 7 green):
Total ways = C(15, 4) = 15! / (4! * (15-4)!) = 1365
Step 2: Calculate the probability of choosing 0 red jelly beans:
Number of ways of choosing 0 red jelly beans = C(7, 4) = 7! / (4! * (7-4)!) = 35
Probability of choosing 0 red jelly beans = 35 / 1365 ≈ 0.0257
Step 3: Calculate the probability of choosing 1 red jelly bean:
Number of ways of choosing 1 red jelly bean = C(8, 1) * C(7, 3) = (8*35) / (1*6) = 280
Probability of choosing 1 red jelly bean = 280 / 1365 ≈ 0.2051
Step 4: Calculate the probability of choosing 2 red jelly beans:
Number of ways of choosing 2 red jelly beans = C(8, 2) * C(7, 2) = (28*21) / (2*1) = 294
Probability of choosing 2 red jelly beans = 294 / 1365 ≈ 0.2150
Step 5: Calculate the probability of choosing 3 red jelly beans:
Number of ways of choosing 3 red jelly beans = C(8, 3) * C(7, 1) = (56*7) / (6*1) = 98
Probability of choosing 3 red jelly beans = 98 / 1365 ≈ 0.0718
Step 6: Calculate the probability of choosing 4 red jelly beans:
Number of ways of choosing 4 red jelly beans = C(8, 4) = 70
Probability of choosing 4 red jelly beans = 70 / 1365 ≈ 0.0513
The probability distribution table for choosing red jelly beans is as follows:
Number of Red Jelly Beans | Probability
--------------------------------------
0 | 0.0257
1 | 0.2051
2 | 0.2150
3 | 0.0718
4 | 0.0513
b) To find the probability that at least 2 chosen jelly beans are green given that they are not all red, we need to calculate the probability of choosing 2, 3, or 4 green jelly beans.
Step 1: Calculate the total number of ways of choosing 4 jelly beans out of 15:
Total ways = C(15, 4) = 15! / (4! * (15-4)!) = 1365
Step 2: Calculate the number of ways of choosing 2 green jelly beans:
Number of ways of choosing 2 green jelly beans = C(7, 2) * C(8, 2) = (21*28) / (2*1) = 294
Step 3: Calculate the number of ways of choosing 3 green jelly beans:
Number of ways of choosing 3 green jelly beans = C(7, 3) * C(8, 1) = (35*8) / (6*1) = 46.67 (approximated as 47)
Step 4: Calculate the number of ways of choosing 4 green jelly beans:
Number of ways of choosing 4 green jelly beans = C(7, 4) = 35
Step 5: Calculate the probability of choosing at least 2 green jelly beans (given that they are not all red):
Probability = (Number of ways of choosing 2 green jelly beans + Number of ways of choosing 3 green jelly beans + Number of ways of choosing 4 green jelly beans) / Total ways
= (294 + 47 + 35) / 1365 ≈ 0.3143
Therefore, the probability that at least 2 chosen jelly beans are green given that they are not all red is approximately 0.3143.