You have two decks of 26 cards. Each card in each of the two decks has a different letter of the alphabet on it. You pick at random one card from each of the two decks. A vowel is worth 3 points and a consonant is worth 0 points. Let X = the sum of the values of the two cards picked. Find E(X), V(X), and the standard deviation of X.
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To find the expected value (E(X)), variance (V(X)), and standard deviation of the sum of the values of the two cards picked, we first need to determine the probability of each possible sum.
1. Calculate the probabilities:
To determine the probability of each possible sum, we first need to find the probability of picking a vowel or a consonant from each deck.
a) Probability of picking a vowel from one deck:
Since there are 26 cards in each deck and 5 vowels in the English alphabet (A, E, I, O, U), the probability of picking a vowel from one deck is 5/26. Therefore, the probability of picking a consonant is 1 - (5/26) = 21/26.
b) Probability of picking a vowel from both decks:
For each deck, the probability of picking a vowel is 5/26. Thus, the probability of picking a vowel from both decks is (5/26) * (5/26) = 25/676.
c) Probability of picking a consonant from both decks:
For each deck, the probability of picking a consonant is 21/26. Hence, the probability of picking a consonant from both decks is (21/26) * (21/26) = 441/676.
d) Probability of picking a vowel from one deck and a consonant from the other:
Since the probability of picking any particular letter is independent of the other, the probability of picking a vowel from one deck and a consonant from the other is (5/26) * (21/26) = 105/676.
2. Calculate the possible sums and their probabilities:
Now, we will determine the possible sums from adding up the values of the two cards picked, as well as their probabilities.
a) Sum of 6 points (two vowels): The probability of this sum is (25/676) * (25/676).
b) Sum of 3 points (vowel/consonant): The probability of this sum is 2 * [(25/676) * (105/676)].
c) Sum of 0 points (two consonants): The probability of this sum is (441/676) * (441/676).
3. Calculate E(X):
E(X) is the sum of the products of each possible value and its corresponding probability. In this case, the possible values are 6, 3, and 0.
E(X) = (6 * (25/676) * (25/676)) + (3 * 2 * (25/676) * (105/676)) + (0 * (441/676) * (441/676))
4. Calculate V(X):
V(X) is the variance of X and is equal to the sum of the squares of the differences between X and E(X), multiplied by their corresponding probabilities.
V(X) = [(6 - E(X))^2 * (25/676)^2] + [(3 - E(X))^2 * 2 * (25/676) * (105/676)] + [(0 - E(X))^2 * (441/676)^2]
5. Calculate the standard deviation:
The standard deviation of X is equal to the square root of V(X).
Standard deviation of X = sqrt(V(X))
By performing the calculations mentioned above, you can find the values of E(X), V(X), and the standard deviation of X.