Suppose a sequence of bridge hands is dealt. Let A be the...

Suppose a sequence of bridge hands is dealt. Let $A$ be the event that each player is dealt one ace on a particular deal. (a) Show that $A$ has a probability of about one-tenth. (Actually 0.1054981993.) (b) What is the probability that one particular player gets no ace for three consecutive deals? (c) Show that the probability that event $A$ occurs at least once in seven deals is about one-half. (Actually, 0.54178581 or 0.5217031 if 0.1 is used as the probability of A.) Hint: By the general multiplication rule (Corollary to Theorem 2.4.1), the number of ways of dealing one bridge hand is $$ \left(\begin{array}{l} 52 \\ 13 \end{array}\right)\left(\begin{array}{l} 39 \\ 13 \end{array}\right)\left(\begin{array}{l} 26 \\ 13 \end{array}\right)=\frac{52 !}{(13 !)^4} . $$

Suppose a sequence of bridge hands is dealt. Let $A$ be the event that each player is dealt one ace on a particular deal.
(a) Show that $A$ has a probability of about one-tenth. (Actually 0.1054981993.)
(b) What is the probability that one particular player gets no ace for three consecutive deals?
(c) Show that the probability that event $A$ occurs at least once in seven deals is about one-half. (Actually, 0.54178581 or 0.5217031 if 0.1 is used as the probability of A.) Hint: By the general multiplication rule (Corollary to Theorem 2.4.1), the number of ways of dealing one bridge hand is
$$
\left(\begin{array}{l}
52 \\
13
\end{array}\right)\left(\begin{array}{l}
39 \\
13
\end{array}\right)\left(\begin{array}{l}
26 \\
13
\end{array}\right)=\frac{52 !}{(13 !)^4} .
$$

Key Concepts

Combinatorial Analysis

This concept involves counting the number of ways outcomes can occur using techniques such as combinations and factorials. It is crucial in determining both the total number of possible outcomes (sample space) and the number of favorable outcomes in probability problems, particularly in contexts where order does not matter, like card dealing.

Multiplication Principle in Combinatorics

The multiplication principle states that if one event can occur in m ways and a subsequent independent event can occur in n ways, then the total number of ways for both events to occur is m × n. This principle is essential when combining sequential independent selections or events to arrive at the total count.

Probability of an Event

This key concept refers to the calculation of the likelihood of an event by taking the ratio of the number of favorable outcomes to the total number of possible outcomes. In combinatorial problems, careful counting via combinations ensures an accurate computation of these probabilities.

Independent Events in Sequential Trials

When events occur independently in a sequence, the outcome of one does not influence the outcomes of the others. The overall probability of a sequence of independent events is the product of their individual probabilities, a principle that is vital when assessing repeated experiments like multiple card deals.

Complementary Probability

The complementary probability rule involves computing the chance that an event occurs by subtracting the probability that it does not occur from one. This approach is particularly useful when it is easier to calculate the probability of the complement, such as determining the likelihood that an event happens at least once over several independent trials.

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