From a deck of bridge cards, two players removed the spade, heart, and club cards numbered 1 to 6 and play with these 18 cards according to the following rules. The player first draws two cards from the deck held by the banker. If the sum of the numbers on the drawn cards is 7, they win; if it is greater than 7, they lose; if it is less than 7, they draw a third card, and only win if the new number completes the sum of the first two to 7. What is the probability that the player wins?
a) First, let's examine the possible outcomes of the game. The player can draw two or three cards in succession, so the possible cases are certain sequences of two or three draws. The fact that the player draws cards randomly from the deck means that the 18 x 17 different outcomes of the first two draws are all equally likely, each with a probability of 1/(18 x 17). This is true regardless of whether the event in question can be further broken down into smaller events (for example, the event of drawing the 2 of hearts and the 3 of spades, which can be broken down into 16 smaller events depending on which of the 16 remaining cards is drawn third), or not (such as drawing the 5 of clubs and the 3 of hearts). It also follows from the randomness that if such a pair of draws can be further broken down into triple draws, then these parts are again equally likely, each with a probability of 1/16 of the probability of the pair of draws (since 16 such triple draws make up a larger event defined by the outcome of the first two draws).

Thus, the event space of the game's outcomes has two types of elementary events: "larger" ones - these have a probability of 1/(18 x 17), and "smaller" ones - these have a probability of 1/(18 x 17 x 16). For the player, there are favorable outcomes among both types, and we will sum their probabilities.

b) The player wins if the sum of the results of the first two draws is 7. The number 7 can be formed in three essentially different ways as the sum of two numbers appearing on the cards:

$$
7=6+1=5+2=4+3
$$

In each of these representations, there are 3 ways to choose the suit of the cards, and 2 ways to choose the order of the cards. Therefore, the number of "large" favorable elementary events is $3 \cdot 3 \cdot 3 \cdot 2$, and the sum of their probabilities is:

$$
p_{1}=\frac{3 \cdot 3 \cdot 3 \cdot 2}{18 \cdot 17}=\frac{3}{17}
$$

c) The number 7 can be broken down into a sum of three terms in four ways:

$$
7=5+1+1=4+2+1=3+3+1=3+2+2
$$

Among these, those with two equal terms correspond to $3 \cdot \binom{3}{2} \cdot 6$ "small" elementary events, because the unique number can be chosen in 3 ways, the two identical numbers in $\binom{3}{2}$ ways, and the order of the three cards in 3! = 6 ways.

There is only one way to break down the number 7 into a sum of three different terms. This corresponds to $3^3 \cdot 6$ "small" elementary events, because each number's suit can be chosen in 3 ways, and the order in 6 ways.

Thus, the sum of the probabilities of the favorable "small" elementary events is

$$
p_{2}=\frac{1}{18 \cdot 17 \cdot 16}\left\{3 \cdot 3 \cdot \binom{3}{2} \cdot 6 + 3^3 \cdot 6\right\}=\frac{2 \cdot 9}{17 \cdot 16}=\frac{9}{136}
$$

d) Therefore, the player wins with a probability of

$$
p=p_{1}+p_{2}=\frac{3}{17}+\frac{9}{136}=\frac{33}{136}=0.243
$$

Csetényi Artúr (Szeged, Radnóti M. Gimn., IV. o. t.)

Remarks. 1. The probability of $3 / 17$ for winning with two cards can also be obtained more simply: whatever the result of the first draw, the player cannot yet be in a losing position, and among the remaining 17 cards, there are always exactly 3 that complete the number on the first card to 7.

2. It may be confusing that the number of all cases is different in parts b) and c). This corresponds to the fact that in the two parts, we did not take the same probability space as a basis. This convenient simplification is permissible if we clearly see that the common space is actually the following: we draw three cards in succession, and the two cases are favorable when the sum of the first two is 7, or the sum of the three cards is 7.