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)
Possible outcomes:
V V ----> (5/26)(5/26) = 25/676
V C ---> (5/26)(21/26) = 105/676
C V --->(21/26)(5/26) = 105/676
C C ---> (21/26)(21/26) = 441/676 , notice the sum is 1
E(x) = 6(25/676) + 3(105/676) + 3(105/676) + 0(441/676) = ....
To find E(X), we need to find the expected value of the sum of the values of the two cards picked.
Let's first determine the probability of picking a vowel or a consonant from each deck.
1. Deck 1:
- There are 26 cards in Deck 1.
- There are 5 vowels (A, E, I, O, U) and 21 consonants in Deck 1.
- The probability of picking a vowel from Deck 1 is 5/26.
- The probability of picking a consonant from Deck 1 is 21/26.
2. Deck 2:
- There are also 26 cards in Deck 2.
- Similarly, there are 5 vowels and 21 consonants in Deck 2.
- The probability of picking a vowel from Deck 2 is also 5/26.
- The probability of picking a consonant from Deck 2 is 21/26.
Now, let's calculate the expected value by considering all possible combinations:
- If a vowel is picked from both decks, the sum of the values would be 3 + 3 = 6.
- If a consonant is picked from both decks, the sum of the values would be 0 + 0 = 0.
- If a vowel is picked from Deck 1 and a consonant is picked from Deck 2, the sum of the values would be 3 + 0 = 3.
- If a consonant is picked from Deck 1 and a vowel is picked from Deck 2, the sum of the values would be 0 + 3 = 3.
Let's calculate the expected value using the probabilities and outcomes:
E(X) = (5/26) * (5/26) * 6 + (21/26) * (21/26) * 0 + (5/26) * (21/26) * 3 + (21/26) * (5/26) * 3
Simplifying:
E(X) = (25/676) * 6 + (441/676) * 0 + (105/676) * 3 + (105/676) * 3
E(X) = (150/676) + (210/676)
E(X) = 360/676
E(X) = 0.5329 (rounded to four decimal places)
Therefore, the expected value of X, E(X), is approximately 0.5329.
To find the expected value E(X) of the sum of values, we need to determine the probability of each possible outcome and multiply it by the corresponding value.
Let's break down the problem step by step:
Step 1: Determine the probability of picking a vowel or consonant from each deck.
- Each deck consists of 26 cards, so the probability of picking a vowel from each deck is 5/26 (since there are 5 vowels in the alphabet).
- Therefore, the probability of picking a consonant from each deck is 21/26 (since there are 21 consonants in the alphabet).
Step 2: Calculate the expected value for each possible combination of cards.
- If we pick a vowel from one deck and a vowel from the other deck, the sum of the values will be 3 + 3 = 6.
- If we pick a vowel from one deck and a consonant from the other deck, the sum of the values will be 3 + 0 = 3.
- If we pick a consonant from one deck and a vowel from the other deck, the sum of the values will be 0 + 3 = 3.
- If we pick a consonant from one deck and a consonant from the other deck, the sum of the values will be 0 + 0 = 0.
Step 3: Calculate the expected value E(X).
- Multiply the probabilities from Step 1 by the corresponding values from Step 2 and sum them up.
(E(X) = (5/26) * (5/26) * 6 + (5/26) * (21/26) * 3 + (21/26) * (5/26) * 3 + (21/26) * (21/26) * 0)
Simplifying this expression, we have:
E(X) = (25/676) * 6 + (105/676) * 3 + (105/676) * 3 + (441/676) * 0
E(X) = 150/676 + 315/676
E(X) = 465/676
Therefore, the expected value E(X) is 465/676.