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) To construct the probability distribution for the value of a 2-card hand, we need to consider all possible combinations of 2 cards from a standard deck of 52 cards.

There are a total of C(52, 2) = 52! / (2! * (52-2)!) = 1326 possible combinations.

Let's calculate the value of each combination:
- If both cards are numbered cards (2-10), the sum of their values will be their total face value.
- If one or both cards are face cards (J, Q, K) or the Ace, their respective values are 10 and 11.

Now, let's count the number of combinations that add up to each value:

Value | Combinations | Probability
-------------------------------------
2 | 1 | 1/1326
3 | 2 | 2/1326
4 | 3 | 3/1326
... | ... | ...
19 | 16 | 16/1326
20 | 15 | 15/1326
21 | 4 | 4/1326

b) The probability of being dealt a hand that adds up to 20 is 15/1326.

Cumulative Probability Distribution Chart:
Value | Combinations | Probability | Cumulative Probability
--------------------------------------------------------------
2 | 1 | 1/1326 | 1/1326
3 | 2 | 2/1326 | 3/1326
4 | 3 | 3/1326 | 6/1326
... | ... | ... | ...
19 | 16 | 16/1326 | 345/1326
20 | 15 | 15/1326 | 360/1326
21 | 4 | 4/1326 | 364/1326

c) The probability of being dealt a 16 or less is the cumulative probability up to 16, which is 345/1326.

d) The probability of being dealt between 12 and 16 (inclusive) is the difference between the cumulative probabilities up to 16 and up to 11.

P(12-16) = P(16) - P(11) = 345/1326 - 105/1326 = 240/1326.

e) The probability of being dealt between 17 and 20 (inclusive) is the difference between the cumulative probabilities up to 20 and up to 16.

P(17-20) = P(20) - P(16) = 360/1326 - 345/1326 = 15/1326.

f) The expected value of a two-card hand can be calculated by multiplying each possible value by its probability and summing them up:

Expected value = (2 * 1/1326) + (3 * 2/1326) + (4 * 3/1326) + ... + (21 * 4/1326)

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 all the possible combinations and their respective probabilities. Here is how we can calculate it:

Step 1: Determine the total number of possible combinations for a 2-card hand. This can be found using the combination formula, which is given by nCr = n! / ((n-r)! * r!), where n is the total number of items and r is the number of items selected.

In this case, we need to find the number of combinations of 52 cards taken 2 at a time. Using the formula, we can calculate it as follows:
C(52,2) = 52! / ((52-2)! * 2!) = 1326

So, there are 1326 possible combinations for a 2-card hand from a standard deck.

Step 2: Determine the probability for each possible outcome.

a) Probability of being dealt a hand that adds up to 21:
To compute this, we need to count the number of combinations that result in a hand totaling 21. There are three possible combinations: (A, 10), (10, A), and (K, A).

There are 4 aces and 16 cards with a value of 10 (4 each of kings, queens, jacks, and 10s) in a standard deck. So, the probability can be calculated as:
P(21) = 3 / 1326 ≈ 0.00226

b) Probability of being dealt a hand that adds up to 20:
To compute this, we need to count the number of combinations that result in a hand totaling 20. There are two possible combinations: (10, 10) and (K, 10) [there are twelve cards with a value of 10, four kings, four queens, and four jacks].

The probability can be calculated as:
P(20) = 2 / 1326 ≈ 0.00151

Step 3: Construct the chart for the cumulative probability distribution.
To construct the cumulative probability distribution, we need to find the probabilities of getting a hand with a value less than or equal to a certain number for all possible values. Here is the chart:

Value | Probability
-------------------
≤ 10 | 0.377
≤ 11 | 0.445
≤ 12 | 0.598
≤ 13 | 0.772
≤ 14 | 0.904
≤ 15 | 0.956
≤ 16 | 0.978
≤ 17 | 0.989
≤ 18 | 0.994
≤ 19 | 0.997
≤ 20 | 0.999
≤ 21 | 1.000

c) Probability of being dealt a 16 or less:
From the cumulative probability distribution chart, we can see that the probability is 0.978.

d) Probability of being dealt between 12 and 16 (inclusive):
To calculate this, we subtract the cumulative probability of getting a hand that sums up to 12 or less from the cumulative probability of getting a hand that sums up to 16 or less. So:
P(12-16) = P(16) - P(12) = 0.978 - 0.598 = 0.380

e) Probability of being dealt between 17 and 20 (inclusive):
To calculate this, we subtract the cumulative probability of getting a hand that sums up to 16 or less from the cumulative probability of getting a hand that sums up to 20 or less. So:
P(17-20) = P(20) - P(16) = 0.999 - 0.978 = 0.021

f) Expected value of a two-card hand:
The expected value can be calculated by summing the products of each possible outcome and its respective probability. Here is the calculation:

Expected Value = (Probability of Value 4 x 4) + (Probability of Value 5 x 5) + ... + (Probability of Value 21 x 21)

Expected Value = (0 x 4) + (0 x 5) + ... + (0.00226 x 21)

Expected Value ≈ 0.04506

Therefore, the expected value of a two-card hand is approximately 0.04506.