Construct the probability distribution for the value of a 2-card hand dealt from a standard deck of 52 cards (all face cards have a value of 10 and an ace has a value of 11). a) What is the probability of being dealt a hand that adds up to 21? b) What is the probability of being dealt a hand that adds up to 20? Construct a chart for the cumulative probability distribution. c) What is the probability of being dealt a 16 or less? d) What is the probability of being dealt between 12 and 16 (inclusive)? e) What is the probability of being dealt between 17 and 20 (inclusive)? f) Find the expected value of a two card hand

a. There are 4 aces and twelve face cards and 4 10s in a deck of 52 cards.

4/52(for aces)*16/51(for cards = 10 once an ace has been drawn)

For probability that all/both are found by multiplying the individual probabilities.

4/52 * 16/51 = ?

b. 16/52 * 15/51 = ?

c. What combinations of two cards will give you 16 or less? Figure combos for 16 = 6,10; 5,9; etc.

It might be easier to calculate for values greater than 16 and subtract from 1.

a= 0.030

b= 0.091

To construct the probability distribution for the value of a 2-card hand, we need to determine the number of possible outcomes for each value and calculate their respective probabilities.

Here is the probability distribution for the value of a 2-card hand:

Value | Number of Outcomes | Probability
-------------------------------------------
2 | 1 | 1/221
3 | 2 | 2/221
4 | 3 | 3/221
5 | 4 | 4/221
6 | 5 | 5/221
7 | 6 | 6/221
8 | 7 | 7/221
9 | 8 | 8/221
10 | 9 | 9/221
11 | 36 | 36/221
12 | 24 | 24/221
13 | 36 | 36/221
14 | 12 | 12/221
15 | 24 | 24/221
16 | 24 | 24/221
17 | 24 | 24/221
18 | 12 | 12/221
19 | 12 | 12/221
20 | 12 | 12/221
21 | 4 | 4/221
Total | 416 | 1

a) The probability of being dealt a hand that adds up to 21 is 4/221.

b) The probability of being dealt a hand that adds up to 20 is 12/221.

Here is a chart for the cumulative probability distribution:

Value | Cumulative Probability
--------------------------------
2 | 1/221
3 | 3/221
4 | 6/221
5 | 10/221
6 | 15/221
7 | 21/221
8 | 28/221
9 | 36/221
10 | 45/221
11 | 81/221
12 | 105/221
13 | 141/221
14 | 153/221
15 | 177/221
16 | 201/221
17 | 225/221
18 | 237/221
19 | 249/221
20 | 261/221
21 | 265/221

c) The probability of being dealt a 16 or less is 201/221.

d) The probability of being dealt between 12 and 16 (inclusive) is (24+24+24+12)/221 = 84/221.

e) The probability of being dealt between 17 and 20 (inclusive) is (24+12+12+12)/221 = 60/221.

f) To find the expected value of a two-card hand, we need to multiply each value by its corresponding probability, and then sum up the products.

Expected Value = (2 * 1/221) + (3 * 2/221) + (4 * 3/221) + ... + (21 * 4/221)

Calculating this sum should give you the expected value.

To construct the probability distribution for the value of a 2-card hand dealt from a standard deck of 52 cards, we need to consider the possible combinations of cards and their corresponding values. Here's how we can calculate the probabilities for each scenario:

Step 1: Determine the number of ways to obtain each value.
- The number of ways to obtain a certain value depends on the number of cards with that value and the number of cards remaining in the deck.
- Each card is equally likely to be drawn from the deck, assuming the deck is well-shuffled.

Now let's calculate the probability for each scenario:

a) Probability of being dealt a hand that adds up to 21:
- There are several combinations that can result in a hand totaling 21:
- Ace and a 10-value card (e.g., King, Queen, or Jack): 4 aces and 16 10-value cards (total of 20 cards).
- Ten, Ten, and an Ace: 16 10-value cards and 4 aces (total of 20 cards).
- The total number of combinations that result in a hand totaling 21 is 20 + 20 = 40.
- The total number of 2-card combinations from a deck of 52 cards is "52 choose 2" = 1,326.
- Therefore, the probability of being dealt a hand that adds up to 21 is 40/1,326 ≈ 0.03 or 3%.

b) Probability of being dealt a hand that adds up to 20:
- There are several combinations that can result in a hand totaling 20:
- Two 10-value cards (e.g., King and Queen): 4 Kings and 4 Queens (total of 8 cards).
- Ten, Ten, and any other card (except Ace): 16 10-value cards and 36 other non-Ace cards (total of 52 cards).
- The total number of combinations that result in a hand totaling 20 is 8 + 52 = 60.
- Therefore, the probability of being dealt a hand that adds up to 20 is 60/1,326 ≈ 0.05 or 5%.

Now let's construct a chart for the cumulative probability distribution:

Value | Number of Ways | Probability
------|---------------|------------
2 | 1 | 0.0769
3 | 2 | 0.1538
4 | 3 | 0.2308
5 | 4 | 0.3077
6 | 5 | 0.3846
7 | 6 | 0.4615
8 | 7 | 0.5385
9 | 8 | 0.6154
10 | 9 | 0.6923
11 | 4 | 0.7692
12 | 4 | 0.8462
13 | 4 | 0.9231
14 | 4 | 1.0000

c) Probability of being dealt a 16 or less:
- From the chart, we can see that the cumulative probability for 16 or less is 0.3846.

d) Probability of being dealt between 12 and 16 (inclusive):
- To find this probability, we can subtract the cumulative probability for 11 from the cumulative probability for 16 or less.
- 1.0000 - 0.7692 = 0.2308

e) Probability of being dealt between 17 and 20 (inclusive):
- To find this probability, we can subtract the cumulative probability for 16 or less from the cumulative probability for 20.
- 0.7692 - 0.6154 = 0.1538

f) Expected value of a two-card hand:
- The expected value is calculated by multiplying each possible value by its corresponding probability, and then summing these products.
- Expected value = (2 * 0.0769) + (3 * 0.1538) + (4 * 0.2308) + ... + (14 * 1.0000)
- Calculating this sum gives an expected value of 9.5385.

Please note that all calculations assume a fair and well-shuffled deck, and the probabilities provided are approximate because we rounded to four decimal places for clarity.