You read in a book about bridge that the probability that...

You read in a book about bridge that the probability that each of the four players is dealt exactly one ace is about $0.11 .$ This means that (a) in every 100 bridge deals, each player has one ace exactly 11 times. (b) in 1 million bridge deals, the number of deals on which each player has one ace will be exactly 110,000 . (c) in a very large number of bridge deals, the percent of deals on which each player has one ace will be very close to $11 \%$ (d) in a very large number of bridge deals, the average number of aces in a hand will be very close to $0.11 .$ (e) If each player gets an ace in only 2 of the first 50 deals, then each player should get an ace in more than $11 \%$ of the next 50 deals.

You read in a book about bridge that the probability that each of the four players is dealt exactly one ace is about $0.11 .$ This means that
(a) in every 100 bridge deals, each player has one ace exactly 11 times.
(b) in 1 million bridge deals, the number of deals on which each player has one ace will be exactly 110,000 .
(c) in a very large number of bridge deals, the percent of deals on which each player has one ace will be very close to $11 \%$
(d) in a very large number of bridge deals, the average number of aces in a hand will be very close to $0.11 .$
(e) If each player gets an ace in only 2 of the first 50 deals, then each player should get an ace in more than $11 \%$ of the next 50 deals.

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Key Concepts

Expected Value versus Observed Frequency

This concept distinguishes between an expected outcome, which is determined by theory and probability calculations, and the actual result observed in a finite number of trials. Expected value provides a long-term average outcome, while actual observations may fluctuate around this value due to natural randomness.

Probability as a Long-run Relative Frequency

This concept means that the probability of an event is interpreted as the proportion of times the event occurs in a very large number of independent trials. It does not guarantee that the event will occur exactly that proportion in any finite sample, but rather that obvious deviations tend to average out as the number of trials increases.

Law of Large Numbers

The Law of Large Numbers states that as the number of repeated trials grows, the sample average (or relative frequency) of an event will tend to get closer to the expected probability of the event. This idea supports the interpretation that, in a very large number of bridge deals, the frequency with which each player is dealt one ace will approach 11%.

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