Question

In a sequential experiment we first flip a fair coin. If head (event H) shows up we roll a fair die and observe the outcome. If tail (event T) shows up, we roll two fair dice. Let X denote the number of sixes that we observe.

a) What is the sample space of X?

b) Find the PMF of X and E[X].

c) Given that X = 1, what is the probability that head showed up in the flip of the coin?

Answer #1

a) as number of six can be 0,1 or 2

therefore sample space S ={0,1,2}

b)

pmf of X:

P(X=0)=P(heads and 0 six on single die)+P(tails and 0 six on 2 dies)=(1/2)*(5/6)+(1/2)*(25/36)=55/72

P(X=1)=P(heads and 1 six on single die)+P(tails and 1 six on 2 dies)=(1/2)*(1/6)+(1/2)*(10/36)=16/72=2/9

P(X=2)=P(tails and 2 six on 2 dies)=(1/2)*(1/36)=1/72

x | f(x) | xP(x) |

0 | 55/72 | 0.000 |

1 | 2/9 | 0.222 |

2 | 1/72 | 0.028 |

total | 0.250 |

from above E(X)=xP(x)=0.25

c)

P(heads|X=1)= P(heads and 1 six on single die)/P(X=1)=(1/2)*(1/6)/(2/9)=9/24=3/8

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