Betty picks 4 random marbles from a bowl containing 3 white, 4 yellow, and 5 blue marbles.
The probability that exactly 1 of the 4 marbles drawn is blue is
_____ (I know that the probability is 5/12 and that simplifies to 0.4166 but that is not one of the answer choices) the answer choices are 0.0707, 0.2135, .3535, and .6465. Can you please help me solve this question I am really stuck and I have tried looking on other websites to no avail.
There are 5 blue out of 12 marbles.
So, there is a 5/12 chance of drawing a blue marble. That means that in 4 draws, the chance of drawing a blue on the 1st draw only is
5/12 * 7/11 * 6/10 * 5/9 = 35/396
Now, add that to the chance of drawing a blue only on the 2nd, 3rd, of 4th draw and you get
4 * 35/396 = 35/99 = 0.3535
To solve this problem, you need to determine the number of favorable outcomes and the total number of possible outcomes.
The total number of possible outcomes can be calculated using combinations. Since Betty is picking 4 marbles from a bowl with a total of 12 marbles, the number of possible outcomes is given by the combination formula:
C(n, r) = n! / (r!(n - r)!)
where n is the total number of marbles and r is the number of marbles Betty is picking.
In this case, n = 12 and r = 4, so the total number of possible outcomes is:
C(12, 4) = 12! / (4!(12 - 4)!) = 495
Now, let's calculate the number of favorable outcomes, which is the number of ways Betty can choose exactly 1 blue marble from the 5 blue marbles and 3 marbles from the remaining 8 marbles (3 white + 4 yellow):
C(5, 1) * C(8, 3) = (5! / (1!(5 - 1)!)) * (8! / (3!(8 - 3)!))
= (5 * 8 * 7 * 6) / (1 * 4 * 3 * 2) * (8 * 7 * 6) / (3 * 2 * 1)
= 70 * 56
= 3920
Now we can calculate the probability of exactly 1 of the 4 marbles drawn being blue by dividing the number of favorable outcomes by the total number of possible outcomes:
P(exactly 1 blue) = favorable outcomes / total outcomes
= 3920 / 495
= 56 / 99
Since none of the answer choices matches the simplified fraction 56/99, it seems that there might be an error in the provided answer choices. Double-check the options or consult your teacher for clarification.
To find the probability that exactly 1 of the 4 marbles drawn is blue, we can use the concept of combinations and the formula for probability.
There are a total of 12 marbles in the bowl, so the number of possible outcomes is given by the combination formula:
C(n, r) = n! / (r!(n-r)!)
where n is the total number of marbles and r is the number of marbles we want to select.
In this case, n = 12 and r = 4, so the total number of possible outcomes is:
C(12, 4) = 12! / (4!(12-4)!) = 12! / (4!8!) = (12 * 11 * 10 * 9) / (4 * 3 * 2 * 1) = 495
Now, let's calculate the number of ways to select 1 blue marble out of the 5 available. We can use the same combination formula:
C(5, 1) = 5! / (1!(5-1)!) = 5
Next, let's calculate the number of ways to select the remaining 3 marbles from the 7 non-blue marbles (3 white and 4 yellow):
C(7, 3) = 7! / (3!(7-3)!) = 7! / (3!4!) = (7 * 6 * 5 * 4) / (4 * 3 * 2 * 1) = 35
Now we can calculate the probability of exactly 1 blue marble:
P(1 blue) = (number of favorable outcomes) / (total number of possible outcomes)
P(1 blue) = (number of ways to select 1 blue marble) * (number of ways to select 3 non-blue marbles) / (total number of possible outcomes)
P(1 blue) = (C(5, 1) * C(7, 3)) / (C(12, 4)) = (5 * 35) / 495 = 175 / 495 = 0.3535
Therefore, the correct answer is 0.3535, which matches one of the answer choices provided.