E(Value of a Card)= ∑ [Value* P(Value)]
To find it, we will first make a table containing the cards and their frequencies in the deck. There are 4 suits of cards, therefore there are 4 cards of each value.
Card Value
Number of Cards With The Same Value (Frequency)
1 (Ace)
4
13 (King)
4
12 (Queen)
4
11 (Jack)
4
10
4
9
4
8
4
7
4
6
4
5
4
4
4
3
4
2
4
Since the number of cards with each value is 4, the probability of drawing a card with each value is equal to 4 divided by the total number of cards, 52.
Card Value
P(Value)
1 (Ace)
4/52
13 (King)
4/52
12 (Queen)
4/52
11 (Jack)
4/52
10
4/52
9
4/52
8
4/52
7
4/52
6
4/52
5
4/52
4
4/52
3
4/52
2
4/52
Now, we can calculate the products of the card values and their probabilities.
Card Value
P(Value)
Value* P(Value)
1 (Ace)
4/52
1*4/52=4/52
13 (King)
4/52
13*4/52=52/52
12 (Queen)
4/52
12*4/52=48/52
11 (Jack)
4/52
11*4/52=44/52
10
4/52
10*4/52=40/52
9
4/52
9*4/52=36/52
8
4/52
8*4/52=32/52
7
4/52
7*4/52=28/52
6
4/52
6*4/52=24/52
5
4/52
5*4/52=20/52
4
4/52
4*4/52=16/52
3
4/52
3*4/52=12/52
2
4/52
2*4/52=8/52
To find the expected card value we need to sum the values from the last column. Let's do it!
E(Value of a Card)= 4/52+52/52+48/52+44/52+40/52+... = 364/52 = 7
The expected value of a card drawn from the deck is 7.
Alternative Solution
Different Approach
The probabilities of drawing cards with different values are the same — they are equal to 452. Let's try to modify the equation for the expected value using this information.
We found that the expected card value is the sum of all the card values multiplied by 452. Let's calculate it!
E(Value of a Card)= 4/52(1+2+3+...+13) = 4/52(91) = 7
b
We are asked to calculate the expected value of the number of spades we get if we are dealt 7 cards, replacing the card each time. First, let's solve something simpler — the expected number of spades if we are dealt only 1 card.
Expected Number of Spades After Drawing 1 Card
We will first find the probabilities of drawing a spade and not a spade. There are 13 spades and 39 cards of other suits in the deck. To find the probabilities of drawing a spade and a card other than spade, we need to divide 13 and 39 by the total number of cards, 52.
Card Suit
Number of Cards
Probability
Spade
13
13/52=1/4
Not Spade
39
39/52=3/4
The amount of spades in hand if we get a spade after drawing one card is 1. The corresponding probability to this event is 14. On the other hand, having 0 spades in hand after drawing one card has a probability of 34.
The Card We Drew
Spades in Hand
Probability
Spade
1
1/4
Not Spade
0
3/4
With this information, we can calculate the expected value of spades in hand after drawing one card!
E(Number of Spades,1Draw)=
1* 1/4+ 0* 34=1/4
Expected Number of Spades After Drawing 7 Cards
Now, let's try to calculate the expected number of spades after we are dealt 7 cards. Notice that since we are dealt cards with replacement, we can think of this event as 7 independent card draws. The expected value of spades after 7 independent card draws is equal to 7 times the expected number of spades after 1 draw.
E(Number of Spades,7Draws)=
7* E(Number of Spades,1Draw)
Earlier, we found that the expected number of spades after 1 draw is 14. Let's substitute that into our equation.
7* E(Number of Spades,1Draw)=
7* 1/4=1.75
The expected value of spades after we are dealt 7 cards with replacement is 1.75. This means that on average we can expect to have around 2 spades in hand after we are dealt 7 cards with replacement.
