A deck of $12$ cards consists of $2$ gold cards and $10$ silver cards. The cards are identical

Question

A deck of $12$ cards consists of $2$ gold cards and $10$ silver cards. The cards are identical other than their colours. In a game, three cards are drawn from the deck of cards at random without replacement. The number of gold cards drawn is denoted by $G$. Explain the significance of stating that the cards are identical other than their colours.[1] Show that $\mathrm{P}(G=2) = \frac{1}{22}$ and find the probabilities of all other possible combinations of the cards drawn.[4] A gold card is worth $4$ points while a silver card is worth $1$ point. The score $X$ is the sum of the points of each of the three drawn cards. Using your answers in part (ii), find the probability distribution of $X$.[2] If the player of the game wins $\$10x$ for a score of $x$ points, what is the expected winnings of the player?[1]

Accepted Answer

(i) Each remaining card is equally likely to be selected. (ii) $P(G=0)=\frac6{11},P(G=1)=\frac9{22},P(G=2)=\frac1{22}$ (last value shown). (iii) $X=3,6,9$ with those respective probabilities. (iv) 45 dollars.

2025 MI P2 Q7

2025 MI P2 Q7

Solution:

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