An urn contains 8 chips, numbered 1 through 8. A sample of 3 is drawn without replacement. What is the probability that the largest chip in the sample is a 5?
Well, let's see. The total number of possible outcomes is the number of ways you can choose 3 chips out of 8, which is "8 choose 3," denoted as C(8, 3). This can be calculated as 8! / (3! * (8-3)!), which simplifies to 56.
Now, out of these 56 possible outcomes, we need to determine how many of them have the largest chip as a 5. The largest chip being a 5 means that we have to choose exactly two chips smaller than 5 and one chip equal to 5.
There are 4 chips smaller than 5 (1, 2, 3, and 4), so we can choose 2 of them in C(4, 2) ways, which is equal to 6. Since we need exactly one chip equal to 5, we're left with just one choice for that.
Therefore, the number of favorable outcomes is 6 * 1 = 6.
So, the probability that the largest chip in the sample is a 5 is the favorable outcomes divided by the total outcomes, which gives us 6/56. Simplifying this fraction, we get 3/28.
So, the probability is 3/28. And just remember, when it comes to clown math, there's always a chuckle hidden somewhere in the numbers.
To calculate the probability of the largest chip in the sample being a 5, we need to determine the total number of possible samples and the number of samples that satisfy the condition.
Step 1: Calculate the total number of possible samples
Since there are 8 chips in total and we are drawing a sample of 3 chips without replacement, the total number of possible samples can be calculated using the combination formula, also known as nCr:
nCr = n! / (r!(n-r)!)
In this case, n = 8 (total number of chips) and r = 3 (number of chips in the sample).
Using the formula:
8C3 = 8! / (3!(8-3)!)
= 8! / (3! * 5!)
= (8 * 7 * 6) / (3 * 2 * 1)
= 56
Therefore, there are 56 possible samples.
Step 2: Calculate the number of samples where the largest chip is a 5
In this case, we have 1 chip numbered 5, and we need to draw another 2 chips from the remaining 7 chips. The largest chip in the sample must be a 5, so there are no other possibilities for the largest chip.
To calculate the number of samples satisfying this condition, we can use the combination formula again:
7C2 = 7! / (2!(7-2)!)
= 7! / (2! * 5!)
= (7 * 6) / (2 * 1)
= 21
Therefore, there are 21 samples where the largest chip is a 5.
Step 3: Calculate the probability
To calculate the probability, divide the number of samples satisfying the condition (21) by the total number of possible samples (56):
Probability = 21 / 56
= 0.375
Therefore, the probability that the largest chip in the sample is a 5 is 0.375 or 37.5%.
To find the probability that the largest chip in the sample is a 5, we need to determine the number of favorable outcomes and the total number of possible outcomes.
First, let's consider the favorable outcomes. In order for the largest chip in the sample to be a 5, we have two cases:
Case 1: The 5 chip is chosen first.
In this case, we need to select two chips out of the remaining 7 chips, which are numbered 1, 2, 3, 4, 6, 7, and 8. The number of ways to do this is given by the combination formula, also known as "n choose k":
C(7, 2) = 7! / (2!(7 - 2)!) = 21
Case 2: The 5 chip is chosen second or third.
In this case, we need to select the 5 chip, along with two other chips from the remaining 7 chips (same set as in Case 1). Again, using the combination formula, we have:
C(7, 2) = 21
Therefore, the total number of favorable outcomes is 21 + 21 = 42.
Now, let's determine the total number of possible outcomes. Since we are drawing 3 chips without replacement, the total number of ways to choose 3 chips out of 8 is given by:
C(8, 3) = 8! / (3!(8 - 3)!) = 56
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
P(largest chip is 5) = favorable outcomes / total outcomes = 42 / 56 = 0.75 or 75%.
Therefore, the probability that the largest chip in the sample is a 5 is 0.75 or 75%.
One chip drawn would have to be 5 and that probability is 1/8. The probability that a second drawn chip is not larger than 5 is 4/7. The probability of the third chip not being larger than 5 is 3/6, or 1/2. This is true regardless of the order they are drawn.
Total probability = (1/8)*(1/2)*(4/7) = 1/28