SOLUTION: A biased six-sided die has one face bearing number 3, two faces number 2 and three faces number 1. A boy tosses the die once and observes the number on the uppermost face. He then

Question 630570: A biased six-sided die has one face bearing number 3, two faces number 2 and three faces number 1. A boy tosses the die once and observes the number on the uppermost face. He then tosses a number of fair coins equal to the number shown on the die. The random variable X is the number of heads obtained.
Construct the probability density function of X.
Answer by Edwin McCravy(20060) (Show Source): You can put this solution on YOUR website!
A biased six-sided die has one face bearing number 3, two faces number 2 and three faces number 1. A boy tosses the die once and observes the number on the uppermost face. He then tosses a number of fair coins equal to the number shown on the die. The random variable X is the number of heads obtained.
Construct the probability density function of X.

  I. P(1 on die) = 3/6 = 1/2
     A. Possible tosses {H,T}
     B. P(1 toss 0 heads) = 1/2
     C. P(1 toss 1 head) = 1/2

 II. P(2 on die) = 2/6 = 1/3
     A. Possible tosses {HH,HT,TH,TT}
     B. P(2 tosses 0 heads) = 1/4
     C. P(2 tosses 1 head)  = 2/4 = 1/2
     D. P(2 tosses 2 heads) = 1/4

III. P(3 on die) = 1/6
     A. Possible tosses {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}
     B. P(3 tosses 0 heads) = 1/8
     C. P(3 tosses 1 head)  = 3/8
     D. P(3 tosses 2 heads) = 3/8 
     E. P(3 tosses 3 heads) = 1/8

P(x=0) = P(D1&H0)+P(D2&H0)+P(D3&H0) = (1/2)(1/2)+(1/3)(1/4)+(1/6)(1/8) = 17/48
P(x=1) = P(D1&H1)+P(D2&H1)+P(D3&H1) = (1/2)(1/2)+(1/3)(1/2)+(1/6)(3/8) = 23/48 
P(x=2) = P(D2&H2)+P(D3&H2) = (1/3)(1/4)+(1/6)(3/8) = 7/48
P(x=3) = P(D3&H3)  = (1/6)(1/8) = 1/48

Probability density function:

  X     P(X)
  0     17/48
  1     23/48
  2      7/48
  3      1/48

Note that the probabilities have sum 48/48 or 1

Edwin

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