13.1 The Basics of Probability Theory Calculate probabilities by counting outcomes in a sample space. Use counting formulas to compute probabilities. Understand how probability theory is used in genetics. Understand the relationship between probability and odds.

Knowing the sample space helps us compute probabilities. Random phenomena – occurrences that vary from day to day, case to case, and situation to situation. Probability – a numerical measure that calculates how a random phenomena will occur in a certain way. Experiment – any observation of random phenomenon. Outcome – the different possible results of the experiment. Sample Space – the set of all possible outcomes for an experiment.

An event is a subset of sample space. In probability theory, an event is a subset of the sample space.  Any subset of the sample space is an event, including the empty set and the whole sample space.

The probability of an event is the sum of the probabilities of the outcomes in that event. Probability of an outcome – a number between 0 and 1 inclusive. The sum of the probabilities of all the outcomes in the sample space must be 1. Probability of an event – written as P(E), is defined as the sum of the probabilities of the outcomes that make up E.

Empirical Assignment of Probabilities – If E is an event and we perform an experiment several times, then we estimated the probability of E as follows: P(E) = the number of times E occurs the number of times the experiment is performed

We can use counting formulas to compute probabilities. We can determine probabilities using empirical information (we make observations and assign probabilities based on those observations). We can also determine probabilities using theoretical information (counting formulas). If E is an event in a sample space S with all equally likely outcomes, then the probability of E is given by the formula: P(E) = n(E) n(S)

If the outcomes in a sample space are not equally likely, then you must add the probabilities of all the individual outcomes in E. Basic Properties of Probability – Assume that S is a sample space for some experiment and E is an event in S < P(E) < 1 2. P(  ) = 03. P(S) = 1

Probability theory helps explain genetic theory. Gregor Mendel experimented with pea plants to show dominant and recessive characteristics.  When he crossed yellow seeded peas with green seeded peas he noticed that each 1 st generation offspring contained one yellow (Y) gene and one green (g) gene. Since the Y gene was dominate, the seeds were all yellow in each offspring.  When he crossed those 1 st generation offspring together he noticed these outcomes which are organized in a Punnett square. Each of the 2 nd generation offspring will have yellow seeds except the ones with the gg pair of genes. These ones will have all green seeds. Y g Y YY Yg g gY gg

In computing odds remember “against” versus “for.” If the outcomes of a sample space are equally likely, then the odds against an event E are simply the number of outcomes that are against E compared with the number of outcomes in favor of E occurring.  We write these odds as n(E`):n(E), where E` is the complement of event E.

There are eight ways for boys and girls to be born in order in a family (bbg bgb bgg gbb gbg ggg bbb ggb). To find the odds against all children being of the same gender, you think of the 6 outcomes that are against this happening versus the 2 outcomes that are in favor. Therefore, the odds are 6:2, which is reduced to 3:1. We could say that the odds in favor of this event are 1:3. The probability of this event is ¼.

Probability formula for computing odds- If E` is the complement of the event E, then the odds against E are: P(E`) P(E)  If the probability of Green Bay winning the Super Bowl is 0.35, what are the odds against them winning? P(G`) = 0.65 = 65 = 13 P(G) = 0.35 = 35 = 7 So, the odds against Green Bay winning are 13 to 7.

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