Rewrite:
Consider a bag containing an unknown number of red and blue balls. Let the number of red balls in the bag be denoted as the realization of a Poisson random variable X with mean 5. Suppose a ball is randomly drawn from the bag with replacement. Find the probability that the drawn ball is red.
Find the probability of drawing a red ball from the bag.
To find the probability of drawing a red ball from the bag, we need to use the concept of a Poisson random variable. In this case, the number of red balls in the bag is given as the realization of a Poisson random variable X with a mean of 5.
The Poisson distribution is a probability distribution that describes the number of events occurring in a fixed interval of time or space, given the average rate of occurrence. In this scenario, the average number of red balls in the bag is 5.
To find the probability of drawing a red ball, we need to calculate the probability mass function (PMF) of the Poisson distribution. The PMF of a Poisson random variable X is given by the formula:
P(X = k) = (e^(-λ) * λ^k) / k!
where λ is the average or mean of the Poisson distribution and k is the number of red balls we are interested in.
In this case, we are interested in the probability of X = 1, which represents drawing a single red ball from the bag.
Let's substitute the values into the formula:
P(X = 1) = (e^(-5) * 5^1) / 1!
Calculating this, we get:
P(X = 1) = (e^(-5) * 5) / 1
Using the value of e ≈ 2.71828, we can evaluate this expression to find the probability of drawing a red ball from the bag.
To find the probability of drawing a red ball from the bag, we are given that the number of red balls in the bag is the realization of a Poisson random variable X with mean 5. This implies that the average number of red balls in the bag is 5.
Since the balls are drawn with replacement, the probability of drawing a red ball on each draw remains constant. We can use the probability mass function of the Poisson distribution to calculate this probability.
The probability mass function (PMF) of a Poisson distribution is given by the formula:
P(X = k) = (e^(-λ) * λ^k) / k!
Where lambda (λ) is the mean of the distribution.
In this case, λ = 5 because the mean number of red balls is 5.
To find the probability of drawing a red ball, we need to calculate P(X = 1), as we are interested in the probability of having exactly 1 red ball.
Let's substitute λ = 5 and k = 1 into the formula:
P(X = 1) = (e^(-5) * 5^1) / 1!
Simplifying further:
P(X = 1) = (e^(-5) * 5) / 1
Using a calculator or software, we can find the value of e^(-5) ≈ 0.00674.
Therefore:
P(X = 1) ≈ (0.00674 * 5) / 1
P(X = 1) ≈ 0.0337
So, the probability of drawing a red ball from the bag is approximately 0.0337 or 3.37%.