#### Probability, Biology tutorial

General Principles:

When coin is tossed it will come up head or tail usually. The result of one toss doesn't influence result of another toss. The ratio of heads or tails would be about 1:1. The results could deviate from ideal 1:1 ratio for any given set of trials. Though, statistical tests show that deviations from expectation are insignificant.

First Law of Probability:

In considering probability of heads or probability of tails we are considering probability of single events. As we have only two possible occurrences when the coin is tossed and both are favorable according to question, answer to question is 1. It is definite that we shall obtain head or tail. Therefore when it is certain that event will occur numerical probability is 1. We can summarize operation by saying that probability of occurrence in one trial, of either of two mutually exclusive events is sum of probabilities of individual occurrence. Events should be mutually exclusive that is both can't occur in one trial, it has to be one or other as in coin toss.

For example we can't come up with 7 when we toss the dice. Hence if numerator in formula is 0 fraction has to be zero. This in effect means that probability of the event occurring can never be less than 0 i.e. it cannot be written with the minus sign. Hence, probability of the occurrence is greater than or equal to 0, and less than or equal to 1 but never greater than 1. In examples of coin toss or dice, probability of getting head is equal to that for tail and same is true for all six sides of die. Though, such equality isn't always true for all possibilities of the event.

Genetic Considerations:

Probabilities of different possibilities depend on kind of event under consideration. For example when two monohybrids are crossed, 3/4 of progeny have dominant phenotypes while ¼ have recessive phenotype. In simple terms then we can say that 3 out of 4 progeny would have recessive. Reason from Punnett squares is that there are 4 possible kinds of fusion between parental gametes to provide progeny. The probability that first offspring of two heterozygotes (monohybrids, Aa x Aa) will have dominant phenotype, answer is ¾ as 3 out of 4 possible gametic fusions will generate dominant phenotypes. One significant feature of the consideration of probabilities is the fact that answer is extremely dependent on phrasing of question. The probability that two monohybrid parents will generate offspring having dominant phenotype is 1 as these parents are potentially able to produce an (at least one) offspring with dominant phenotype. The reasons for the probability that first child of Aa x aa parents will have th recessive phenotype, is 1/2 are:

i) A particular child is indicated, and

ii) There are only two possible genotypes and phenotypes, Aa and aa.

From cross of two monohybrid parents, 3/4 of other progeny have dominant phenotype. Though, such progeny can be either AA or Aa in genotype. Among progeny with dominant phenotype, two genotypes are present in the ratio of 1:2 respectively.

The probability that if farmer put his hand in the bowl having yellow seeds from cross of Yy x Yy, he would select a seed which is YY in genotype is:

The phrasing of question eradicates green seeds from number of possible occurrences to be considered. Relevant seeds occur in the ratio of 1 : 2, thus, answer is 1/3. Probability of Yy would be 2/3; so again sum of probabilities for this series of events containing only yellow progeny is (1/3 + 2/3) equal to 1. It is significant to remember that sum of all probabilities is one even if you have re-adjusted total number of possibilities? Probability of occurrences of either of two mutually exclusive events in one trial is sum of probabilities of their individual occurrences.

Second Law of Probability:

If two coins are tossed at the same time, appearance of the head or tail on one coin doesn't in any way influence what appears on other coin. Therefore we have given possible combinations: HH, HT, TH and TT. All such combinations are evenly probable, assuming coins weren't biased. Using probability formula, probability that both coins would appear heads in the single toss is 1/2. Same would be true for both coins appearing tails. Though, answer with respect to situation where coins show up differently will depend on question asked. For instance, find probability that one coin will appear head and other tail? According to possible combinations listed earlier, there are 2 favorable occurrences (combinations) that satisfy condition HT and TH, hence answer is 2/4 i.e. 1/2. Different question is: find probability that first coin will be head and second tail. In this question answer is 1/4 as we have specified what we expect of each coin, making only HT combination the favorable occurrence.

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