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Maruf Karaev
Maruf Karaev
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In the basket lies m black balls and n reds. Vasya takes out a random ball from the basket and, if it is black, then replaces it with red, and if it is red, then puts it back. Find the mathematical expectation and variance of the number of red balls in the basket after k iterations of this procedure. Both answers must be compact expressions (i.e. do not contain summation signs, ellipses, etc.).

I tried to solve this problem in this way. Let X - number of red balls after k iterations. Easy to see that n <= X <= n + m

We should find

$\sum_{i=n}^{n+m} P(X=i) * i$

$P(X = n) = (\frac{n}{n+m}) ^ {k}$$P(X = i) = (\frac{n}{n+m}) ^ {i}$ But couldn't find other probabilities. Because the probability of taking black/red ball changes after we take a black ball. How I can solve this ?

In the basket lies m black balls and n reds. Vasya takes out a random ball from the basket and, if it is black, then replaces it with red, and if it is red, then puts it back. Find the mathematical expectation and variance of the number of red balls in the basket after k iterations of this procedure. Both answers must be compact expressions (i.e. do not contain summation signs, ellipses, etc.).

I tried to solve this problem in this way. Let X - number of red balls after k iterations. Easy to see that n <= X <= n + m

We should find

$\sum_{i=n}^{n+m} P(X=i) * i$

$P(X = n) = (\frac{n}{n+m}) ^ {k}$ But couldn't find other probabilities. Because the probability of taking black/red ball changes after we take a black ball. How I can solve this ?

In the basket lies m black balls and n reds. Vasya takes out a random ball from the basket and, if it is black, then replaces it with red, and if it is red, then puts it back. Find the mathematical expectation and variance of the number of red balls in the basket after k iterations of this procedure. Both answers must be compact expressions (i.e. do not contain summation signs, ellipses, etc.).

I tried to solve this problem in this way. Let X - number of red balls after k iterations. Easy to see that n <= X <= n + m

We should find

$\sum_{i=n}^{n+m} P(X=i) * i$

$P(X = i) = (\frac{n}{n+m}) ^ {i}$ But couldn't find other probabilities. Because the probability of taking black/red ball changes after we take a black ball. How I can solve this ?

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Maruf Karaev
Maruf Karaev
  • 11
  • 1

In the basket lies m black balls and n reds. Vasya takes out a random ball from the basket and, if it is black, then replaces it with red, and if it is red, then puts it back. Find the mathematical expectation and variance of the number of red balls in the basket after k iterations of this procedure. Both answers must be compact expressions (i.e. do not contain summation signs, ellipses, etc.).

I tried to solve this problem in this way. Let X - number of red balls after k iterations. Easy to see that n <= X <= n + m

We should find

$\sum_{i=n}^{n+m} P(X=i) * i$

$P(X = n) = (\frac{n}{n+m}) ^ {k}$ But couldn't find other probabilities. Because the probability of taking black/red ball changes after we take a black ball. How I can solve this ?

In the basket lies m black balls and n reds. Vasya takes out a random ball from the basket and, if it is black, then replaces it with red, and if it is red, then puts it back. Find the mathematical expectation and variance of the number of red balls in the basket after k iterations of this procedure. Both answers must be compact expressions (i.e. do not contain summation signs, ellipses, etc.).

In the basket lies m black balls and n reds. Vasya takes out a random ball from the basket and, if it is black, then replaces it with red, and if it is red, then puts it back. Find the mathematical expectation and variance of the number of red balls in the basket after k iterations of this procedure. Both answers must be compact expressions (i.e. do not contain summation signs, ellipses, etc.).

I tried to solve this problem in this way. Let X - number of red balls after k iterations. Easy to see that n <= X <= n + m

We should find

$\sum_{i=n}^{n+m} P(X=i) * i$

$P(X = n) = (\frac{n}{n+m}) ^ {k}$ But couldn't find other probabilities. Because the probability of taking black/red ball changes after we take a black ball. How I can solve this ?

Source Link
Maruf Karaev
Maruf Karaev
  • 11
  • 1

mathematical expectation after k iterations

In the basket lies m black balls and n reds. Vasya takes out a random ball from the basket and, if it is black, then replaces it with red, and if it is red, then puts it back. Find the mathematical expectation and variance of the number of red balls in the basket after k iterations of this procedure. Both answers must be compact expressions (i.e. do not contain summation signs, ellipses, etc.).