Question

find the probability distribution of number of kings obtained when 2 cards are drawn at random from 52 cards

Answer 1

To find the probability distribution of the number of kings when 2 cards are drawn, we calculate the probabilities for drawing 0, 1, and 2 kings separately. These probabilities account for the changing composition of the deck after each draw without replacement.

Step-by-step explanation:

To find the probability distribution of the number of kings obtained when 2 cards are drawn at random from a deck of 52 cards without replacement, we consider the following possible outcomes:

• 0 kings
• 1 king
• 2 kings

Let's calculate the probabilities for each outcome step by step:

1. The probability of drawing 0 kings: Since there are 4 kings in the deck, the probability of not drawing a king on the first draw is 48/52. Without replacement, the deck now has 51 cards, and the probability of not drawing a king on the second draw is 47/51. So, the combined probability is (48/52) * (47/51).
2. The probability of drawing 1 king: The probability of drawing a king on the first draw is 4/52. Without replacement, there are now 48 non-kings left, and the probability of drawing a non-king on the second draw is 48/51. Alternatively, the probability of drawing a non-king first (48/52) followed by a king (4/51) must also be considered. These probabilities are additive: (4/52) * (48/51) + (4/51) * (48/52).
3. The probability of drawing 2 kings: The probability of drawing a king on the first draw is 4/52, and with one king gone, the probability of drawing another is 3/51. The combined probability is (4/52) * (3/51).

Thus, the probability distribution for the number of kings drawn is:

• P(X = 0 kings) = (48/52) * (47/51)
• P(X = 1 king) = (4/52) * (48/51) + (4/51) * (48/52)
• P(X = 2 kings) = (4/52) * (3/51)

Answer 2

The probability distribution for the number of kings obtained when drawing 2 cards from a standard deck is:
`\( P(X=0) = (24 * 47)/(26 * 51) \), \( P(X=1) = (64)/(221) \), \( P(X=2) = (1)/(221) \).`

To find the probability distribution of the number of kings obtained when 2 cards are drawn at random from a standard deck of 52 cards, we can consider all possible outcomes and calculate their probabilities.

There are four kings in a deck, and we are drawing 2 cards without replacement. Let's define the random variable X as the number of kings obtained. The possible values for X are 0, 1, and 2.

1. Probability of getting 0 kings (no kings):

There are 48 non-king cards left after the first draw, and 47 non-king cards left after the second draw. So, the probability of getting 0 kings is:

`\[ P(X=0) = (48)/(52) * (47)/(51) \]`

2. Probability of getting 1 king:

We can get exactly one king in two ways: either the first card is a king and the second is not, or the first card is not a king and the second is. The probability of getting 1 king is the sum of these probabilities:

`\[ P(X=1) = (4)/(52) * (48)/(51) + (48)/(52) * (4)/(51) \]`

3. Probability of getting 2 kings:

The probability of getting 2 kings is when both cards drawn are kings:

`\[ P(X=2) = (4)/(52) * (3)/(51) \]`

Now, you can compute these probabilities:

`\[ P(X=0) = (48)/(52) * (47)/(51) \]\[ P(X=1) = (4)/(52) * (48)/(51) + (48)/(52) * (4)/(51) \]\[ P(X=2) = (4)/(52) * (3)/(51) \]`

Simplify each expression to obtain the numerical values.

Let's simplify the expressions:

1. Probability of getting 0 kings
`(\(P(X=0)\))`:

`\[ P(X=0) = (48)/(52) * (47)/(51) = (24)/(26) * (47)/(51) = (24 * 47)/(26 * 51) \]`

2. Probability of getting 1 king
`(\(P(X=1)\))`:

`\[ P(X=1) = (4)/(52) * (48)/(51) + (48)/(52) * (4)/(51) = (1)/(13) * (16)/(17) + (12)/(13) * (4)/(17) \]`

Combine the fractions:
`\[ P(X=1) = (16 + 48)/(13 * 17) = (64)/(221) \]`

3. Probability of getting 2 kings
`(\(P(X=2)\))`:

`\[ P(X=2) = (4)/(52) * (3)/(51) = (1)/(13) * (1)/(17) = (1)/(221) \]`

So, the final probability distribution is:

`\[ P(X=0) = (24 * 47)/(26 * 51) \]\[ P(X=1) = (64)/(221) \]\[ P(X=2) = (1)/(221) \]`