: Four cards are drawn without replacement from a well shuffled standard deck of cards. Let X be the number of aces drawn.
12. What are the possible values of the random variable X? Write them in increasing order as a set using roster (or list) notation.
13. Sketch a histogram for the probability distribution of the variable X.
14. What is the expected value of X?
Here are the answers I came up with but I would like to know if they are correct. If they are correct, can you direct me on how to plot the histogram.
P(x=0)= 0.7187
P(x=1)= .6388
P(x=2)= .0041
P(x=3)=.0001773
P(x=4)=.0000003737
I know some or even all of your answers must be wrong, since the sum of all the prob's has to add up to 1
The sum of your first 2 results is already over 1
first one is correct,
I assume you did
C(48,4) /C(52,4) = .718736..
Doing the 2nd the same way
you want to chose 1 ace from the 4 aces, then 3 of the remaining 48 non-aces
= C(4,1) x C(48,3) / C(52,4) = .2556
3rd:
2 aces, 2 non-aces
prob(x=2) = C(4,2) x C(48,2)/C(52,4) = .0250
complete the remaining two, using the same procedure
Check to see if the sum of all results = 1
The possible values of the random variable X, representing the number of aces drawn, are:
X = {0, 1, 2, 3, 4}
To plot the histogram for the probability distribution of X, you would need to label the x-axis with the values of X (0, 1, 2, 3, 4) and the y-axis with the corresponding probabilities (0.7187, 0.6388, 0.0041, 0.0001773, 0.0000003737). Each bar on the histogram would represent the probability of that specific value occurring.
To construct the histogram, you can draw bars of different heights on the y-axis corresponding to the probabilities. The x-axis would have bars at positions 0, 1, 2, 3, and 4. The height of each bar would be proportional to its respective probability.
To determine the possible values of the random variable X, we need to consider the number of aces that can be drawn from the deck. Since there are only 4 aces in a standard deck, the maximum number of aces you can draw from 4 cards is 4. However, it is also possible to not draw any aces, resulting in a minimum value of 0.
Therefore, the possible values of X are {0, 1, 2, 3, 4}.
To plot a histogram for the probability distribution of the variable X, you need to consider the probabilities associated with each value of X. Here are the correct probabilities for each value of X:
P(x=0)= 0.7187
P(x=1)= 0.2388
P(x=2)= 0.0413
P(x=3)= 0.0018
P(x=4)= 0.0004
To plot the histogram, you can follow these steps using a graphing software or tool:
1. Create a horizontal axis labeled "Number of Aces Drawn (X)".
2. Create a vertical axis labeled "Probability".
3. Mark the values 0, 1, 2, 3, and 4 on the horizontal axis.
4. For each value of X, draw a rectangle above the corresponding value on the horizontal axis.
- The height of the rectangle represents the probability for that value of X.
- The width of the rectangle can be the same for each value of X.
5. Label the rectangles with their respective probabilities.
The finished plot should show bars representing the probabilities for each value of X, with the height of each bar indicating the associated probability.