Question

1. In the game Merfergamon, there’s a 22% chance of being dealt a Lone, a 32% chance of being dealt a Dual, and a 27% chance of being dealt a Triple. These are the lowest valued hands. Round to four decimal places if necessary.

a)  What's the probability of being dealt something better?

b) What's the probability of being dealt either a Lone or a Dual in your first hand of cards?

c) What's the probability of not being dealt a Triple?

d) Over 6 deals, assuming the cards are shuffled well enough for separate deals to be independent, what is the probability of getting dealt at least one hand with a Dual?  

Answer 1

P (Event = X) = 1 - P(Event = not X)

For two independent events X and Y, P( Event = X and Event = Y) = P(Event = X)*P(Event = Y)

P(Event = X or Event = Y) = P(Event = X) + P(Event = Y) - P(Event = X and Event = Y)

P(Being dealt a Lone) = 22/100 = 0.22

P (Being dealt a Dual) = 32/100 = 0.32

P(Being dealt a triple) = 27/100 = 0.27

a) P(Being dealt something better) = 1 - P(Being dealt one of the hands)

P(Being dealt one of the hands) = P(Being dealt a Lone) + P (Being dealt a Dual) + P(Being dealt a triple) - P(Being dealt a Lone and Dual and Triple)

{Since these are the lowest value cards}

{Since, only one hand can be dealt at a time, P(Being dealt a Lone and Dual and Triple) = 0, the events are mutually exclusive}

P(Being dealt one of the hands) = 0.22 + 0.32 + 0.27 = 0.81

Therefore, P(Being dealt something better) = 1 - 0.81 = 0.19.

b) P(Being dealt a Lone or a Dual in the first hand of cards) = P(Being dealt a Lone in the first hand of cards) + P(Being dealt a Dual in the first hand of cards) - P(Being dealt a Lone and a Dual in the first hand of cards)

{Since, First hand of cards can either have a Lone or a Dual, P(Being dealt a Lone and a Dual in the first hand of cards) = 0 i.e. both the events are mutually exclusive}

P(Being dealt a Lone or a Dual in the first hand of cards) = 0.22 + 0.32 = 0.54.

c) P(Not being dealt a Triple) = 1 - P(Being dealt a Triple) = 1 - 0.27 = 0.73.

d) P(Being dealt at least one hand with a Dual over 6 Deals) = 1 - P(Not being dealt a Dual over 6 Deals)

P(Not being dealt a Dual over 6 Deals) = P(Not Being dealt a dual in the First deal and Not Being dealt a dual in the Second deal and Not Being dealt a dual in the Third deal and Not Being dealt a dual in the Fourth deal and Not Being dealt a dual in the Fifth deal and Not Being dealt a dual in the Sixth deal)

= P(Not Being dealt a dual in the First deal)*P(Not Being dealt a dual in the Second deal)*P(Not Being dealt a dual in the Third deal)*P(Not Being dealt a dual in the Fourth deal)*P(Not Being dealt a dual in the Fifth deal)*P(Not Being dealt a dual in the Sixth deal)

{Since P(Not being dealt a Dual in a deal) = 1 - P(Being dealt Dual in a deal) = 1 - 0.32 = 0.68}

P(Not being dealt a Dual over 6 Deals) = (0.68)*(0.68)*(0.68)*(0.68)*(0.68)*(0.68) = 0.68^6 = 0.098867482.

P(Being dealt at least one hand with a Dual over 6 Deals) = 1 - 0.68^6 = 0.901132517.

Do comment for any doubts.