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Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of number of red balls. Also, Find the mean of the random variable. 

Find out the probability distribution and mean if the Two balls are drawn at random one by one with replacements from an urn containing equal number of red balls and green balls.

by Maivizhi A

Updated Mar 07, 2024

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Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of number of red balls. Also, Find the mean of the random variable.

Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of number of red balls. Also, Find the mean of the random variable. 

The mean of the random variable is 1.

  • (Green, Green):
    Probability = (x/2x) * (x/2x) = 1/4 (all green)
  • (Red, Green) or (Green, Red):
    Probability = (x/2x) * (x/2x) = 1/4 (one red, one green - order doesn't matter) 
  • (Red, Red):
    Probability = (x/2x) * (x/2x) = 1/4 (all red)

 So, the probability of getting one red ball is 2 * (1/4) = 1/2.

Probability Distribution:

Let X be the number of red balls drawn. The probability distribution of X is:

  • P(X = 0) = 1/4 (no red balls)
  • P(X = 1) = 1/2 (one red ball)
  • P(X = 2) = 1/4 (two red balls)

Mean (Expected Value):

  • E(X) = (0 * P(X = 0)) + (1 * P(X = 1)) + (2 * P(X = 2))
  • E(X) = (0 * 1/4) + (1 * 1/2) + (2 * 1/4)
  • E(X) = 0 + 1/2 + 1/2
  • E(X) = 1

The probability distribution of the number of red balls is:

  • P(0 red balls) = 1/4
  • P(1 red ball) = 1/2
  • P(2 red balls) = 1/4 

So, the mean (expected value) is 1.

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Example Problems to Practice 

  1. A bag contains 5 red marbles and 7 blue marbles. Two marbles are drawn at random with replacement. Find the probability distribution of the number of red marbles drawn.
  2. A box contains 4 white socks and 3 black socks. Two socks are drawn at random without replacement. Find the probability distribution of the number of white socks drawn.
  3. A deck of 52 playing cards is shuffled and two cards are drawn at random without replacement. Find the probability distribution of the number of face cards (jacks, queens, and kings) drawn.

Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of number of red balls - FAQs

1. What are the colors of the ball contained in the urn?

Red balls and Green Balls

2. The probability distribution of number of red balls is ------

The probability distribution of number of red balls is 1.

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