I need help with a probability quesition that has been presented in my grade 12 class. I'm really not sure how to go about solving it.
"What is the probability of being dealth exactly 3 of a kind in your 5-card poker hand?"
Thanks for any help.
the probability is probably 60% OR 3/5
I found this on another site. I didn't verify the calculations, but it does look similar to results I've had to calculate before.
Prob(Three of a kind) =
(13C1 x 4C3 x 12C2 x 4^2)/ 52C5 = .02112845
Explanation:
13C1 is number of ways of choosing the face value for the three cards.
4C3 is the number of ways of choosing the suits for these three cards.
12C2 is the number of ways of choosing the face values for the other
two cards. (order does NOT matter)
4^2 is the way the suits for these last two cards can be chosen.
52C5 = (52 choose 5) is the number of unrestricted ways that 5 cards
can be selected from the 52 in the pack.
1. Can a network have exactly one vertex with an odd degree? Provide evidence to support your answer.
2.Explain why for any matrix A,(At)t = A. You man use exampels to illustrate your answer.
Some1 owns a choc. shop. A popular product is a box of 40chocs. 5 of which are nuts.
if u pick 2 out, randomly, What is the probability that both chocolates are nuts
i neeed help for mode and meamn i need help my homework
A jury pool has 17 men and 22 women, from which 12 will be selected. Assuming that each person is equally likely to be selected and that the jury is selected at random, find the probability the jury consists of
a) all women.
b) 8 women and 4 men
1. Can a network have exactly one vertex with an odd degree? Provide evidence to support your answer.
Yes, a network can have exactly one vertex with an odd degree. This is because in a network, the number of edges connected to a vertex determines its degree. If a network has exactly one vertex with an odd degree, all other vertices must have an even degree in order for the sum of all degrees to be even. The evidence for this is that it satisfies the Handshaking Lemma, which states that the sum of the degrees of all vertices in a network is always even.
2. Explain why for any matrix A,(At)t = A. You may use examples to illustrate your answer.
For any matrix A, the transpose of A, denoted as At, is obtained by interchanging the rows and columns of A. Taking the transpose twice, (At)t, would result in the original matrix A. This is because the first transpose swaps the rows and columns, and the second transpose swaps them back to their original order.
To illustrate this, let's consider a simple example:
A = [1 2]
[3 4]
The transpose of A, At, would be:
At = [1 3]
[2 4]
Taking the transpose again, (At)t, gives us the original matrix, A:
(At)t = [(1 3)t]
[(2 4)t]
(At)t = [1 2]
[3 4]
Therefore, for any matrix A, (At)t is equal to A.
As for your probability questions, I'd be happy to help as well. Here are the answers:
3. If you randomly pick 2 chocolates from a box of 40, where 5 of them are nuts, the probability of getting both chocolates as nuts can be calculated as follows:
Probability = (Number of ways to select 2 nut chocolates) / (Total number of ways to select 2 chocolates)
Number of ways to select 2 nut chocolates = (5C2) = 10
Total number of ways to select 2 chocolates = (40C2) = 780
Probability = 10 / 780 = 1 / 78
Therefore, the probability of picking both chocolates as nuts is 1/78.
4. For the mean and mode, please provide specific details about your homework question so that I can assist you accordingly.
To solve the probability problem, we can use the combination formula.
1. What is the probability of being dealt exactly 3 of a kind in your 5-card poker hand?
The formula to calculate this probability is:
Prob(Three of a kind) = (13C1 × 4C3 × 12C2 × 4^2) / 52C5
Explanation:
- 13C1 is the number of ways of choosing the face value for the three cards (one of each kind).
- 4C3 is the number of ways of choosing the suits for these three cards.
- 12C2 is the number of ways of choosing the face values for the other two cards (order does not matter).
- 4^2 is the ways the suits for these last two cards can be chosen.
- 52C5 = (52 choose 5) is the number of unrestricted ways that 5 cards can be selected from the 52-card deck.
By plugging in these values into the formula, you can calculate the probability. According to the calculation from the provided formula, the probability of being dealt exactly 3 of a kind in your 5-card poker hand is approximately 0.0211 or 2.11%.
2. Can a network have exactly one vertex with an odd degree? Provide evidence to support your answer.
Yes, a network can have exactly one vertex with an odd degree. This can be proven by considering the Handshaking Lemma in Graph Theory. The Handshaking Lemma states that the sum of degrees of all vertices in a graph equals twice the number of edges in the graph. Since the sum of degrees must be even (twice an integer), it is possible to have a network with one vertex having an odd degree, as long as the sum of all other vertices' degrees is even.
3. Explain why for any matrix A, (At)t = A. You may use examples to illustrate your answer.
To show why (At)t = A for any matrix A, we can use the property of matrix transposition and the fact that transposing a matrix twice returns the original matrix.
By definition:
- (At)t is the transpose of the transpose of matrix A.
- Transposing matrix A involves interchanging its rows and columns.
When A is transposed, the rows become columns and the columns become rows. So, (At)t means taking the transpose of the transpose of A, which effectively means switching the rows and columns twice, resulting in the original matrix A.
For example, consider the matrix A:
A = [1 2 3; 4 5 6]
Taking the transpose of A gives:
At = [1 4; 2 5; 3 6]
Taking the transpose of At gives:
(At)t = [1 2 3; 4 5 6]
As you can see, (At)t is equal to A.
4. If someone owns a chocolate shop and there is a box of 40 chocolates, 5 of which are nuts, what is the probability of randomly picking 2 chocolates and having both of them be nuts?
To calculate the probability, we need to consider the total number of possible outcomes and the number of favorable outcomes.
Total number of possible outcomes:
There are 40 chocolates in the box, so the total number of possible outcomes is 40C2 (40 choose 2), which is the number of ways to choose 2 chocolates from a group of 40.
Number of favorable outcomes:
There are 5 nut chocolates in the box, so the number of favorable outcomes is 5C2 (5 choose 2), which is the number of ways to choose 2 chocolates from a group of 5.
Now, we can calculate the probability:
Probability = Favorable Outcomes / Total Outcomes
Probability = 5C2 / 40C2
Using the combination formula, we can calculate the probabilities.
5C2 = (5!)/(2!(5-2)!) = 10
40C2 = (40!)/(2!(40-2)!) = 780
Probability = 10/780 ≈ 0.0128 or 1.28%
5. Mode and Mean:
The mode is the value(s) that occur(s) most frequently in a dataset. It represents the highest frequency.
The mean, also known as the average, is calculated by summing all the values in a dataset and dividing the sum by the number of values.
To calculate the mode:
- Sort the dataset in ascending order.
- Count the frequency of each value in the dataset.
- Identify the value(s) with the highest frequency as the mode(s).
To calculate the mean:
- Sum all the values in the dataset.
- Divide the sum by the total number of values.
If you have a specific dataset or more information, I can help you calculate the mode and mean.
6. A jury pool consists of 17 men and 22 women. From this pool, 12 individuals will be selected at random to form a jury. We need to find the probability that:
a) The jury consists of all women.
b) The jury consists of 8 women and 4 men.
a) To find the probability that the jury consists of all women, we need to calculate the number of possible juries that can be formed with only women and divide it by the total number of possible juries.
Number of possible juries with all women = 22C12 (22 choose 12), which is the number of ways to select 12 women from a pool of 22.
Total number of possible juries = (17 + 22)C12, which is the number of ways to select 12 individuals from a pool of 17 men and 22 women.
Probability of an all-women jury = Number of possible juries with all women / Total number of possible juries
b) To find the probability that the jury consists of 8 women and 4 men, we need to calculate the number of possible juries that can be formed with 8 women and 4 men and divide it by the total number of possible juries.
Number of possible juries with 8 women and 4 men = (22C8) * (17C4), which is the number of ways to choose 8 women from 22 and 4 men from 17.
Total number of possible juries = (17 + 22)C12, which is the number of ways to select 12 individuals from a pool of 17 men and 22 women.
Probability of 8 women and 4 men jury = Number of possible juries with 8 women and 4 men / Total number of possible juries