In a group of 54 candidates, 16 are female and the remaining are male. If four are to be randomly selected, what is the probability that there will be only female candidates selected?
17 coins and 1 six-sided number cube are tossed together. What is the probability of getting 17 heads or a four?
Please explain!!
I think I figured out the second answer.... 1/12 ??
(1/2)17+(1/6)=
Since a coin has 2 sides, the probability of landing a heads is 1/2, and because there are 17 of them, you multiply the 1/2 by 17. The die is being rolled once so the porbability of rolling a 4 is 1/6. Add them together you get what you need.
To solve both of these probability questions, we need to use the concept of combinations and calculate the ratio of favorable outcomes to total outcomes.
First, let's solve the problem with the group of candidates.
1. Probability of selecting only female candidates:
Given that there are 16 female candidates in a group of 54, the total number of combinations for selecting 4 candidates is calculated using the combination formula: C(n, k) = n! / (k!(n-k)!), where n is the total number of candidates and k is the number of candidates to be selected.
So, we need to calculate C(54, 4) which equals 54! / (4!(54-4)!) = 54! / (4!50!) = (54 * 53 * 52 * 51) / (4 * 3 * 2 * 1) = 13,685,376.
Now, out of the total combinations, the number of combinations with only female candidates is C(16, 4) = 16! / (4!(16-4)!) = 16! / (4!12!) = (16 * 15 * 14 * 13) / (4 * 3 * 2 * 1) = 3,920.
Therefore, the probability of selecting only female candidates is:
P(only female) = favorable outcomes / total outcomes = 3,920 / 13,685,376 ≈ 0.00029.
Now, let's move on to the second problem involving coins and a six-sided number cube.
2. Probability of getting 17 heads or a four:
When tossing 17 coins, the total number of combinations is 2^17, as each coin has two possible outcomes: heads or tails.
The six-sided number cube has 6 possible outcomes: 1, 2, 3, 4, 5, or 6.
The favorable outcomes occur when you get either 17 heads or a four, which means the number of outcomes with 17 heads plus the number of outcomes with a four need to be counted.
- Number of outcomes with 17 heads:
There is only one way to get 17 heads when tossing 17 coins.
- Number of outcomes with a four:
When tossing the number cube, the probability of getting a four is 1/6, so the number of outcomes with a four is 1.
Therefore, the total number of favorable outcomes is 1 + 1 = 2.
The probability of getting 17 heads or a four is:
P(17 heads or four) = favorable outcomes / total outcomes = 2 / (2^17) ≈ 2 / 131,072 ≈ 0.0000153.
That's how you can solve both of these probability problems using the concept of combinations and favorable outcomes divided by total outcomes.