# Coin toss probability with random variables

A coin is tossed 3 times. If the same side falls for all of the three times, the coin gets tossed once more. Describe the space of elementary events and find the CDF of the vector $(X, Y)$, where $X$ is the number of fallen tails and $Y$ is the number of tosses.

I have tried calculating the probabilities for $X=0, Y=0$, $X=1, Y=0$, and so on.. but haven't gotten the right result. Hints on how to approach this problem would be great, thanks.

• Have you tried drawing a tree-representation of that experiment ? – Evargalo May 16 '17 at 16:46
• Since you have only 10 results, 6 with proba $1/8$ and 4 with proba $1/16$, it is not so hard to denumbrate all the cases. – Evargalo May 16 '17 at 16:48

Cases:

• P(HHHH) = 1/16
• P(HHHT) = 1/16
• P(TTTH) = 1/16
• P(TTTT) = 1/16
• P(HHT) = 1/8
• P(HTH) = 1/8
• P(HTT) = 1/8
• P(THH) = 1/8
• P(THT) = 1/8
• P(TTH) = 1/8

Calculate the probability of each such case then sum for each group having a given vector of final counts:

• P(4H, 0T) = 1/16
• P(0H, 4T) = 1/16
• P(3H, 1T) = 1/16
• P(1H, 3T) = 1/16
• P(2H, 1T) = 3/8
• P(1H, 2T) = 3/8