# If both games cost \$1 to pay? getting two heads and two tails on four coins wins you \$3. or you get \$2 for every six that appears when three standard dice are rolled?

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2015-08-20T19:57:35+05:30

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1.   to play : 1 \$ each game
2.    2 H + 2 T  gets 3 \$    on four coins game
3.    3 dice are rolled.  Every 6 , gets  \$2.

we have to decide which game is more winning ????

game 1:
number of outcomes when four coins are tossed once :  2^4 = 16
Number of outcomes that are favourable = 6
( HHTT, HTHT, HTTH, TTHH, THTH, THHT  )  = 4! /(2 * 2)
arrangements of 4 letters, with 2 being identical and another 2 being identical.

now  Probability of winning = 6/16 = 0.375
Expected winnings =  0.375 * \$3 =  \$ 1.125

game 2:

number of out comes:  6^3 = 216
favourable out comes =
1.  outcomes with 1 six only :  1 * 5 * 5 + 5 * 1 * 5 + 5 * 5 * 1 = 75
2. outcomes with 2 sixes :   1 * 1 * 5 + 1 * 5 * 1 + 1 * 1 * 5 = 15
3. outcomes with 3 sixes :   1 * 1 * 1 = 1

expected winnings =  Sum of P( \$2) * \$2 + P(\$4) * \$4 + P(\$6) * \$6 + P(\$0) * \$ 0
= ( 75/216)  * \$2  + (15/216) * \$4 + (1/216) * \$6
= \$ 1

So  Game 1 is better to play as the expected winnings are more.

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