# Probability Theory, work check

• WWCY
In summary, the probability of drawing at least 2 kings given that there is at least one king is 0.1237.

## Homework Statement

Hi all, could someone give my working a quick skim to see if it checks out? Many thanks in advance.

Suppose that 5 cards are dealt from a 52-card deck. What is the probability of drawing at least two kings given that there is at least one king?

## The Attempt at a Solution

Let ##B## denote the event that at least 2 kings are drawn, and ##A## the event that at least 1 king is drawn. Because ##B## is a strict subset of ##A##,
$$P(B|A) = P(A \cap B)/P(A) = P(B)/P(A)$$
Compute ##P(A)##, ##P(A^c )## denotes the probability of not drawing a single king.
$$P(A) = 1 - P(A^c) = 1 - \frac{48 \choose 5}{52 \choose 5} \approx 1 - 0.6588$$
Compute ##P(B)##, ##P(B^c)## denotes the probability of not drawing at least 2 kings, which is the sum of probabilities of drawing 1 king ##P(1)## and the probability of not drawing a single king ##P(A^c)##.
$$P(B) = 1 - P(B^c) = 1 - (P(1) - P(A^c))$$
$$P(1) = \frac{5 \times {4\choose 1} \times 48 \times 47 \times 46 \times 45 }{52 \times 51 \times 50 \times 49 \times 48} \approx 0.299$$
where the numerator is the number of ways one can have a hand of 5 containing a single king.
$$P(B) \approx 1 - (0.299 + 0.6588) \approx 0.0422$$
finally,
$$P(B|A) = P(B)/P(A) = 0.0422 / (1-0.6588) \approx 0.1237$$

I'm pretty sure that there is a quicker way to do all of this even if my work checks out, I'd appreciate if someone could demonstrate a more efficient calculation!

I agree with how you have calculated it. I get 0.1222.

PeroK
I get the same answer as @verty.

The way I looked at it was:

## p = \frac{1 - p_0 - p_1}{1-p_0} = 1 - \frac{p_1}{1-p_0}##

Where ##p_n## is The probability of drawing exactly ##n## kings.

WWCY
WWCY said:

## Homework Statement

Hi all, could someone give my working a quick skim to see if it checks out? Many thanks in advance.

Suppose that 5 cards are dealt from a 52-card deck. What is the probability of drawing at least two kings given that there is at least one king?

## The Attempt at a Solution

Let ##B## denote the event that at least 2 kings are drawn, and ##A## the event that at least 1 king is drawn. Because ##B## is a strict subset of ##A##,
$$P(B|A) = P(A \cap B)/P(A) = P(B)/P(A)$$
Compute ##P(A)##, ##P(A^c )## denotes the probability of not drawing a single king.
$$P(A) = 1 - P(A^c) = 1 - \frac{48 \choose 5}{52 \choose 5} \approx 1 - 0.6588$$
Compute ##P(B)##, ##P(B^c)## denotes the probability of not drawing at least 2 kings, which is the sum of probabilities of drawing 1 king ##P(1)## and the probability of not drawing a single king ##P(A^c)##.
$$P(B) = 1 - P(B^c) = 1 - (P(1) - P(A^c))$$
$$P(1) = \frac{5 \times {4\choose 1} \times 48 \times 47 \times 46 \times 45 }{52 \times 51 \times 50 \times 49 \times 48} \approx 0.299$$
where the numerator is the number of ways one can have a hand of 5 containing a single king.
$$P(B) \approx 1 - (0.299 + 0.6588) \approx 0.0422$$
finally,
$$P(B|A) = P(B)/P(A) = 0.0422 / (1-0.6588) \approx 0.1237$$

I'm pretty sure that there is a quicker way to do all of this even if my work checks out, I'd appreciate if someone could demonstrate a more efficient calculation!

Your method is OK. The method of Perok in #3 is faster. However, I have one quibble: you ought to keep more significant figures when doing calculations that involve subtractions, so as to avoid "subtractive error magnification". In your case you do not do too badly, getting 0.1237 instead of 2257/18472 ≈ 0.12218, but the general principle still holds. (In some problems subtractive error magnification can lead to huge errors, perhaps even incorrect final signs, etc.)

Ray Vickson said:
Your method is OK. The method of Perok in #3 is faster. However, I have one quibble: you ought to keep more significant figures when doing calculations that involve subtractions, so as to avoid "subtractive error magnification". In your case you do not do too badly, getting 0.1237 instead of 2257/18472 ≈ 0.12218, but the general principle still holds. (In some problems subtractive error magnification can lead to huge errors, perhaps even incorrect final signs, etc.)

Ah okay, I'll keep that in mind the next time. Thanks!

## 1. What is probability theory?

Probability theory is a branch of mathematics that deals with the study of random phenomena. It is used to model and analyze uncertain events and to make predictions about the likelihood of those events occurring.

## 2. How is probability theory used in real life?

Probability theory is used in a variety of fields, including finance, insurance, science, and engineering. It is used to make predictions and informed decisions in situations where there is uncertainty, such as in weather forecasting, risk assessment, and stock market analysis.

## 3. What is the difference between probability and statistics?

Probability and statistics are closely related, but they have different focuses. Probability deals with the theoretical study of random phenomena and the likelihood of events occurring, while statistics deals with the collection, analysis, and interpretation of data to make inferences and predictions about a population.

## 4. Can probability theory be applied to all situations?

Probability theory can be applied to many situations, but it is important to note that it is based on assumptions and simplifications. It may not accurately model complex real-world scenarios, but it can still provide valuable insights and predictions.

## 5. How can I improve my understanding of probability theory?

To improve your understanding of probability theory, it is important to practice solving problems and working with different types of probability distributions. It may also be helpful to study the fundamental principles and concepts of probability theory, such as independence, conditional probability, and Bayes' theorem.