There are 4 cardio stations and six weight lifting stations at the gym. You decide to chose 4 stations randomly for today's workout. Whats the probability that you choose all weight lifting or all cardio?
I assume you never use the same machine twice
first cardio = 4/10 = 2/5
second = 3/9 =1/3
third = 2/8 = 1/4
fourth = 1/7
product = 2/5 *1/3*1/4*1/7
= 1/210
now weight
first =6/10 = 3/5
second = 5/9
third = 4/8 = 1/2
fourth = 3/7
product = 3/5 * 5/9 * 1/2 * 3/7
=1/14
so
1/210 + 1/14 =1/210 + 15/210
= 16/210
= 8/105 = .0762
To find the probability of choosing all weight lifting or all cardio stations, we first need to determine the total number of ways to choose 4 stations out of the 10 available. This can be calculated using the combination formula, often denoted as "nCr."
The total number of ways to choose 4 stations out of 10 is calculated as:
10C4 = 10! / (4! * (10-4)!) = 10! / (4! * 6!) = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1) = 210.
Now, let's calculate the probability of choosing all weight lifting stations. Since the six weight lifting stations are the only ones we want to choose, we need to find the number of ways to choose 4 stations from the 6 weight lifting stations, i.e., 6C4.
6C4 = 6! / (4! * (6-4)!) = 6! / (4! * 2!) = (6 * 5 * 4 * 3) / (4 * 3 * 2 * 1) = 15.
To find the probability, we divide the number of favorable outcomes (15) by the total number of possible outcomes (210):
Probability of choosing all weight lifting = 15 / 210 = 1 / 14 ≈ 0.0714.
Similarly, we can calculate the probability of choosing all cardio stations using the same logic.
Probability of choosing all cardio = 4C4 / 10C4 = 1 / 210 ≈ 0.0048.
Therefore, the probability of choosing all weight lifting or all cardio stations is the sum of these two probabilities:
Probability of choosing all weight lifting or all cardio = Probability of choosing all weight lifting + Probability of choosing all cardio
= 1/14 + 1/210 = 16/210 ≈ 0.0762.
So, the probability of choosing all weight lifting or all cardio stations is approximately 0.0762 or 7.62%.
To find the probability of choosing all weightlifting or all cardio stations, we need to calculate the probability of each scenario separately and then add them together.
First, let's calculate the probability of choosing all weightlifting stations. Since there are 6 weightlifting stations and we are choosing 4 stations randomly, we can use the combination formula to calculate the number of ways we can choose 4 weightlifting stations out of 6:
C(6, 4) = 6! / (4! * (6-4)!) = 6! / (4! * 2!) = (6 * 5 * 4 * 3) / (4 * 3 * 2 * 1) = 15
Now, let's calculate the total number of ways we can choose 4 stations out of the total 10 stations (4 cardio + 6 weightlifting):
C(10, 4) = 10! / (4! * (10-4)!) = 10! / (4! * 6!) = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1) = 210
So, the probability of choosing all weightlifting stations is 15/210.
Next, let's calculate the probability of choosing all cardio stations. Since there are 4 cardio stations and we are choosing 4 stations randomly, we can use the combination formula:
C(4, 4) = 4! / (4! * (4-4)!) = 4! / (4! * 0!) = 1
So, the probability of choosing all cardio stations is 1/210.
Finally, let's add the probabilities of both scenarios together:
15/210 + 1/210 = 16/210
Therefore, the probability of choosing all weightlifting or all cardio stations is 16/210, which can be simplified to 8/105.