Suppose you draw a card from a deck (with replacement) 10 times in a row. What is the probability that you get exactly 3 hearts? Round your answer to 3 decimal places.
To find the probability of getting exactly 3 hearts when drawing a card from a deck with replacement 10 times in a row, we'll use the binomial probability formula.
The probability of getting exactly k successes (3 hearts in this case) in n independent Bernoulli trials (10 draws) is given by the formula:
P(X=k) = (n C k) * p^k * (1-p)^(n-k)
Where:
- n is the total number of trials (10 draws)
- k is the number of successful trials (3 hearts)
- (n C k) is the binomial coefficient, also known as the number of combinations of n items taken k at a time
- p is the probability of a single successful trial (getting a heart)
- (1-p) is the probability of a single unsuccessful trial (not getting a heart)
In a standard deck of cards, there are 52 cards and 13 hearts. Therefore, the probability of getting a heart in a single draw is p = 13/52 = 1/4.
Plugging the values into the formula, we have:
P(X=3) = (10 C 3) * (1/4)^3 * (3/4)^(10-3)
Let's calculate this using Python:
```python
from math import comb
k = 3
n = 10
p = 1/4
probability = comb(n, k) * p**k * (1-p)**(n-k)
round(probability, 3)
```
Running this code, we find that the probability of getting exactly 3 hearts is approximately 0.250.
Therefore, the probability is approximately 0.250 (rounded to 3 decimal places).
To find the probability of getting exactly 3 hearts when drawing a card from a deck with replacement 10 times in a row, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.
In this problem, there are 4 hearts in a standard deck of 52 cards.
Let's break down the problem into multiple steps:
Step 1: Calculate the probability of picking a heart in a single draw.
Since there are 4 hearts in a deck of 52 cards, the probability of picking a heart is 4/52, which simplifies to 1/13.
Step 2: Calculate the probability of not picking a heart in a single draw.
Since there are 52 total cards and 4 are hearts, there are 52 - 4 = 48 non-heart cards. Therefore, the probability of not picking a heart is 48/52, which simplifies to 12/13.
Step 3: Calculate the probability of getting exactly 3 hearts in 10 draws.
Since each draw is independent, we can use the binomial probability formula. The formula for the probability of getting exactly k successes in n trials is:
P(k successes in n trials) = (nCk) * (p^k) * (q^(n-k))
Where:
- nCk is the combination (binomial coefficient) of n items taken k at a time.
- p is the probability of success in a single trial.
- q is the probability of failure in a single trial (1 - p).
Plugging in the values for this problem:
n = 10 (10 draws)
k = 3 (exactly 3 hearts in the 10 draws)
p = 1/13 (probability of picking a heart)
q = 12/13 (probability of not picking a heart)
P(3 hearts in 10 draws) = (10C3) * ((1/13)^3) * ((12/13)^(10-3))
Step 4: Calculate the combination (binomial coefficient) 10C3.
The combination formula is given by:
nCk = n! / (k! * (n-k)!)
Using this formula, we can calculate 10C3:
10C3 = 10! / (3! * (10-3)!)
10C3 = 10! / (3! * 7!)
10C3 = (10 * 9 * 8) / (3 * 2 * 1)
10C3 = 10 * 3 * 4
10C3 = 120
Step 5: Calculate the final probability.
P(3 hearts in 10 draws) = (120) * ((1/13)^3) * ((12/13)^(10-3))
P(3 hearts in 10 draws) = 120 * (1/2197) * (12/169)
P(3 hearts in 10 draws) ≈ 0.044
Therefore, the probability of getting exactly 3 hearts when drawing a card from a deck with replacement 10 times in a row is approximately 0.044 (rounded to 3 decimal places).