(1 pt) A box contains 25 yellow, 28 green and 38 red jelly beans.

If 13 jelly beans are selected at random, what is the probability that:
a) 5 are yellow?
b) 5 are yellow and 7 are green?
c) At least one is yellow?

In total, there are 25+28+38=91 jelly beans.

Using (n,r) to stand for "n choose r"
=n!/(r!(n-r)!)

To choose y yellow, g green and r red jelly beans, the number of ways to choose is given by
(25,y)*(28,g)*(38,r)
and the number of ways to choose the same number of jelly beans irrespective of colour is
(25+28+38,y+g+r)=(91,y+g+r)

So the probability is:
(25,y)*(28,g)*(38,r) / (91,y+g+r)

For 5 yellow, we not worry about the other two colours, so they can be combined, call it x. We need 8 of x.
Probability
=(25,5)*(66,8)/(91,13)
=53130*5743572120/1917283000904460
=0.1592 (approx.)

For (b), it is a similar expression. I get approx. 0.0012

For (c), you would choose 13 out of green and red, and subtract from the probability of 1.
1-(66,13)(25,0)/(91,13)
=1-0.01
=0.99 (approximately)

See also following link for a detailed explanation:
http://mathforum.org/library/drmath/view/69151.html

a) The probability that 5 jelly beans selected at random are yellow can be calculated by dividing the number of favorable outcomes (selecting 5 yellow jelly beans) by the number of total outcomes (selecting any 13 jelly beans).

The number of ways to choose 5 yellow jelly beans out of 25 is given by the combination formula C(25, 5) = 25! / (5!(25-5)!).

Similarly, the number of ways to select any 13 jelly beans out of the total 91 jelly beans is C(91, 13) = 91! / (13!(91-13)!).

Therefore, the probability is:
P(5 yellow) = C(25, 5) / C(91, 13)

b) The probability that 5 jelly beans are yellow and 7 are green can be calculated by dividing the number of favorable outcomes (selecting 5 yellow and 7 green jelly beans) by the number of total outcomes (selecting any 13 jelly beans).

The number of ways to choose 5 yellow jelly beans out of 25 is given by the combination formula C(25, 5) = 25! / (5!(25-5)!).

Similarly, the number of ways to choose 7 green jelly beans out of 28 is C(28, 7) = 28! / (7!(28-7)!).

Therefore, the probability is:
P(5 yellow and 7 green) = (C(25, 5) * C(28, 7)) / C(91, 13)

c) To calculate the probability of at least one yellow jelly bean being selected, we need to consider the complementary probability of selecting no yellow jelly beans.

The number of ways to choose 13 jelly beans with no yellow jelly beans out of the 63 non-yellow jelly beans (green + red) is C(63, 13) = 63! / (13!(63-13)!).

Therefore, the probability of selecting at least one yellow jelly bean is:
P(at least one yellow) = 1 - C(63, 13) / C(91, 13)

However, as a Clown Bot, I don't clown around with probabilities. Let's just say there's a good chance you'll end up with a colorful assortment of flavors! 🤡🍬

To find the probability, we need to first determine the total number of possible outcomes and the number of favorable outcomes for each scenario.

a) To find the probability that 5 jelly beans are yellow, we need to choose 5 jelly beans from the total of 25 yellow jelly beans. The remaining 8 jelly beans can be of any color. So, the total number of possible outcomes is the combination of 13 jelly beans taken 13 at a time:

Total possible outcomes = 63 C 13

The number of favorable outcomes is the combination of 5 yellow jelly beans taken 5 at a time multiplied by the combination of 8 non-yellow jelly beans taken 8 at a time:

Favorable outcomes = (25 C 5) * (38 C 8)

Therefore, the probability that 5 jelly beans are yellow is:

Probability = (Favorable outcomes) / (Total possible outcomes)

b) To find the probability that 5 jelly beans are yellow and 7 are green, we need to choose 5 yellow jelly beans from the total of 25 yellow jelly beans and 7 green jelly beans from the total of 28 green jelly beans. The remaining jelly beans can be of any color. So, the total number of possible outcomes is the combination of 13 jelly beans taken 13 at a time:

Total possible outcomes = 63 C 13

The number of favorable outcomes is the combination of 5 yellow jelly beans taken 5 at a time multiplied by the combination of 7 green jelly beans taken 7 at a time multiplied by the combination of 1 non-yellow and non-green jelly bean taken 1 at a time:

Favorable outcomes = (25 C 5) * (38 C 8) * (28 C 7) * (1 C 1)

Therefore, the probability that 5 jelly beans are yellow and 7 are green is:

Probability = (Favorable outcomes) / (Total possible outcomes)

c) To find the probability that at least one jelly bean is yellow, we need to find the probability of the complement event, which is the probability that no jelly bean is yellow. The number of favorable outcomes in this case is choosing all the jelly beans from the non-yellow jelly beans:

Favorable outcomes = (38 C 13)

Therefore, the probability that at least one jelly bean is yellow is:

Probability = 1 - (Favorable outcomes) / (Total possible outcomes)

To find the probabilities, we need to use the concept of probability and combinations.

a) To calculate the probability that 5 jelly beans are yellow, we need to find the number of ways we can select 5 yellow jelly beans out of the total possible combinations when 13 jelly beans are selected randomly.

First, let's calculate the total possible combinations of selecting 13 jelly beans from the given box. Since there are a total of 25 yellow, 28 green, and 38 red jelly beans in the box, there is a total of 25 + 28 + 38 = 91 jelly beans.

Using the combination formula, the total number of combinations is given by:

C(n, r) = n! / (r! * (n - r)!)

where n is the total number of items, and r is the number of items to be selected.

So, the total possible combinations when selecting 13 jelly beans is:

C(91, 13) = 91! / (13! * (91 - 13)!)

Next, we need to calculate the number of ways we can select 5 yellow jelly beans from the 25 yellow jelly beans in the box:

C(25, 5) = 25! / (5! * (25 - 5)!)

Now, to find the probability that exactly 5 jelly beans are yellow, we divide the number of ways to get 5 yellow jelly beans by the total number of possible combinations:

P(5 yellow) = C(25, 5) / C(91, 13)

b) To calculate the probability that 5 jelly beans are yellow and 7 are green, we need to find the number of ways we can select 5 yellow jelly beans from the 25 yellow jelly beans and 7 green jelly beans from the 28 green jelly beans in the box:

C(25, 5) = 25! / (5! * (25 - 5)!)
C(28, 7) = 28! / (7! * (28 - 7)!)

The probability is then given by:

P(5 yellow and 7 green) = (C(25, 5) * C(28, 7)) / C(91, 13)

c) To calculate the probability that at least one jelly bean is yellow, we can find the complement of the probability that no jelly beans are yellow. In other words, we calculate the probability that all 13 jelly beans are either green or red, and subtract it from 1.

To find the probability that all 13 jelly beans are either green or red, we calculate the number of ways to select 13 jelly beans from the 28 green jelly beans and 38 red jelly beans:

C(28, 13) = 28! / (13! * (28 - 13)!)
C(38, 13) = 38! / (13! * (38 - 13)!)

The probability that at least one jelly bean is yellow is:

P(at least one yellow) = 1 - (C(28, 13) + C(38, 13)) / C(91, 13)

Now you can substitute the values into the formulas to calculate the probabilities.