In a certain carnival game the player selects two balls at random from an urn containing 3 red balls and 9 white balls. The player receives $4 if he draws two red balls and $1 if he draws one red ball. He loses $2 if no red balls are in the sample. Determine the probability distribution for the experiment of playing the game and observing the player's earnings. The probability to draw two red balls is __, to draw one red ball is __, and to draw zero red balls is __.

Answer:The probability to draw two red balls = 1/22The probability to draw one red ball = 9/22The probability to draw no red ball = 12/22Step-by-step explanation:Number of Red balls = 3Number of White balls = 9If the player draws two red balls, he receives $4If the player draws one red ball, he receives $1If the player draws no red ball, he looses $2The total number of balls = 3+9= 12Let R represent Red ballsLet W represent White ballsThe probability that the player earns $4 by picking two red balls is represented as Pr(R1 n R2) Pr(R1 n R2) = Pr(R1) * Pr(R2)Pr(R1) = 3/12 = 1/4Pr(R2) = 2/11(we assume he draws without replacement) Pr(R1 n R2) = 1/4*2/11 = 2/44 = 1/22The probability of earning $4 is 1/22The probability of drawing one red ball is Pr(R1 n W2) or Pr(W1 n R2) Pr(R1) = 3/12 = 1/4Pr(W2) = 9/11Pr(W1) = 9/12 = 3/4Pr(R2) = 3/11Pr(R1 n W2) or Pr(W1 n R2) = (1/4 * 9/11) + (3/4 * 3/11)= (9/44) + (9/44)= 18/44= 9/22Therefore, the probability of earning $1 is 9/22The probability that no red ball is chosen is Pr(W1nW2)Pr(W1) = 9/12 = 3/4Pr(W2) = 8/12Pr(W1nW2) = 3/4 * 8/11 = 24/44 = 12/22therefore. the probability of loosing $2 is 12/22