EXAMPLE 2 Tossing Two Coins: The Importance of Sample Space

Probabilities can be hard to determine without detailing or diagramming the sample space. For example, E. P. Northrop notes that even the great 18th-century French mathematician Jean le Rond d’Alembert tripped on the question “In two coin tosses, what is the probability that heads will appear at least once?” Because the number of heads could be 0, 1, or 2, d’Alembert reasoned (incorrectly) that each of those possibilities would have an equal probability of , and so he reached the (wrong) answer of .

What went wrong? Well, could not be the fully detailed sample space because “1 head” can happen in more than one way. For example, if you flip a dime and a penny once apiece, you could display the sample space with a table, such as the one in Figure 8.5.

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Figure 8.5: Figure 8.5 A table illustrating the outcomes of flipping two coins. As you can see, Figure 8.5 is a table with 2 rows and 2 columns, which displays outcomes: .
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Figure 8.6: Figure 8.6 Tree diagram illustrating outcomes of flipping two coins.

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Another way to generate these four outcomes is with the tree diagram shown in Figure 8.6. Each possible left-to-right pathway through the branches generates an outcome. For example, going up (to dime “heads”) and then down (to penny “tails”) yields the outcome HT.

Either way, we can see that the sample space has 4, not 3, equally likely outcomes. With the table or tree diagram in view, you may already see that the correct probability of at least 1 head is not , but .