662 CHAPTER 11 Probability Table 11.2 Second-Generation Offspring Dominant Trait Number with Dominant Trait Recessive Trait Number with Recessive Trait Ratio of Dominant to Recessive P (Dominant Trait) Yellow seeds 6022 Green seeds 2001 3.01 to 1 ≈ 6022 8023 0.75 Round seeds 5474 Wrinkled seeds 1850 2.96 to 1 ≈ 5474 7324 0.75 From his work, Mendel concluded that the sex cells (now called gametes) of the true yellow (dominant) pea plant carried some factor that caused the offspring to be yellow and that the gametes of the green variety had a variant factor that “induced the development of green plants.” In 1909, Danish geneticist W. Johannsen called these factors “genes.” Mendel’s work led to the understanding that each pea plant contains two genes for color, one that comes from the mother and the other from the father. If the two genes are alike—for instance, both for yellow plants or both for green plants— the plant will be that color. If the genes for color are different, the plant will grow the color of the dominant gene. Thus, if one parent contributes a gene for the plant to be yellow (dominant) and the other parent contributes a gene for the plant to be green (recessive), the plant will be yellow. Theoretical Probability Recall that the possible results of an experiment are called outcomes. When you roll a die, the possible outcomes are 1, 2, 3, 4, 5, and 6. If the die is a fair die it is equally likely that you will roll any one of the six possible numbers. Definition: Equally Likely Outcomes If each outcome of an experiment has the same chance of occurring as any other outcome, we say that the outcomes are equally likely outcomes . When we toss a fair coin, two outcomes are possible: heads or tails. Since the chance of tossing a head is the same as tossing a tail, the results of this experiment are equally likely outcomes. When we draw a single card from a standard deck of cards, there are 52 different cards that can be drawn. Since the chance of drawing any one card is the same for all 52 cards, the results of this experiment are equally likely outcomes. If an experiment has equally likely outcomes , the theoretical probability of event E, symbolized by P E( ), may be determined with the following formula. Theoretical Probability The theoretical probability of an event, P E( ), can be determined using the following formula. = P E E ( ) number of outcomes favorable to total number of possible outcomes Did You Know? Batting Averages m Jeff McNeil If Jeff McNeil of the New York Mets gets three hits in his first three at bats of the season, he is batting a thousand (1.000). Over the course of the 162 games of the season (with three or four at bats per game), however, his batting average will fall closer to 0.326 (his 2022 major league leading batting average). In 2022, out of 533 at bats, McNeil had 174 hits, an average above all other players’ but much less than 1.000. His batting average is a relative frequency (or empirical probability) of hits to at bats. It is only the long-term average that we take seriously because it is based on the law of large numbers. ERIK S LESSER/ EPA-EFE/Shutterstock
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