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It is believed that one of the reasons Mendel's ideas remained misunderstood for over three decades was the mathematical reasoning they contained.

Mendel assumed that gamete formation followed the laws of probability with respect to the distribution of factors.

## Probability Basics

Probability is the chance that an event has to occur between two or more possible events. For example, when we flip a coin, what is the chance that it will fall face-up? And in a 52-card deck, what is the chance of a diamond card being drawn?

## Random Events

Events such as getting a face when tossing a coin, drawing an ace of diamonds from the deck, or getting a face 6 when throwing a die are called random events (from Latin). *alea*, luck) because each of them has the same chance of occurring in relation to their respective alternate events.

Following are the probabilities of occurrence of some random events. Try to explain why each occurs with the indicated probability.

- The probability of drawing a spade card from a 52-card deck is ¼
- The probability of drawing any king from a 52-card deck is 1/13.
- The odds of drawing the king of spades from a 52-card deck are 1/52.

The formation of a certain type of gamete with another allele of a pair of genes is also a random event. A heterozygous individual **Aa** has the same probability of forming allele-bearing gametes **THE** than forming gametes with the allele **a (1/2 A: 1/2 a).**

## Independent Events

When the occurrence of one event does not affect the probability of occurrence of another, one speaks of independent events. For example, when tossing multiple currencies at the same time, or the same currency multiple times in a row, one result does not interfere with the others. Therefore, each result is an event independent of the other.

Similarly, the birth of a child with a given phenotype is an independent event in relation to the birth of other children of the same couple. For example, imagine a couple who have had two sons; how likely is a third child to be female? Since each child's upbringing is an independent event, the chance of a girl being born, assuming men and women are born as often, is 1/2 or 50%, as with any birth.

## The "e" rule

Probability theory says that the probability of two or more independent events occurring together is equal to the product of the probabilities of occurring separately. This principle is popularly known as the "e" rule because it answers the question: **What is the probability of an event occurring And another simultaneously?**

Suppose you toss a coin twice. How likely are you to get two "guys", that is, "guy" in the first release and "guy" in the second? The chance of "face" on the first play is, as we have seen, equal to ½; The chance of "face" on the second play is also equal to 1/2. Thus the probability of these two events occurring together is 1/2 X 1/2 = 1/4.

In the simultaneous roll of three dice, what is the probability of drawing "face 6" in all? The chance of “face 6” on each die is equal to 1/6. Therefore the probability of “face 6” occurring in the three dice is 1/6 X 1/6 X 1/6 = 1/216. This means that obtaining three simultaneous “6 faces” will repeat on average 1 every 216 moves.

A couple wants to have two children and wants to know the likelihood that they are both male. Assuming that the probability of being male or female is ½, the probability of a couple having two boys is 1/2 X 1/2, that is, ¼.

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## The "or" rule

Another probability principle says that the occurrence of two mutually exclusive events is equal to the sum of the probabilities with which each event occurs. This principle is popularly known as the "or" rule because it answers the question: What is the probability of an OR event occurring?

For example, the probability of getting heads or tails when tossing a coin is equal to 1 because it represents the probability of heads being added to the probability of tails (1/2 + 1/2 = 1). To calculate the probability of getting “face 1” or “face 6” when rolling a die, simply add the odds of each event: 1/6 + 1/6 = 2/6.

In certain cases we need to apply both the "and" rule and the "or" rule in our probability calculations. For example, when flipping two coins, how likely are you to get heads on one coin and crown on the other? To occur heads in the first coin **AND** “Crown” on Monday, **OR** “Crown” on the first and “dude” on the second. So in this case the "e" rule applies combined with the "or" rule. The probability of occurring “face” AND “crown” (1/2 X 1/2 = 1/4) OR “crown” and “face” (1/2 X 1/2 = 1/4) is equal to **1/2** (1/4 + 1/4).

**The same reasoning applies to the problems of genetics.** For example, how likely is a couple to have two children, one male and one female? As we have already seen, the probability of a child being male is ½ and of being female is also ½. There are two ways for a couple to have a boy and a girl: the first child to be a boy AND the second child to be a girl (1/2 X 1/2 = 1/4) OR the first to be a girl and the second to be a boy (1 / 2 X 1/2 = 1/4). The final probability is 1/4 + 1/4 = 2/4, or 1/2.