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Twelve Equally Likely Outcomes of Rolling a Die and Flipping a Penny
Rolling Die Flipping Penny
D 1 P H
D 1 P T
D 2 P H
D 2 P T
D 3 P H
D 3 P T
D 4 P H
D 4 P T
D 5 P H
D 5 P T
D 6 P H
D 6 P T

Of the 12 possible outcomes, the die has a 2/12 (or 1/6) probability of rolling a two, and the penny has a 6/12 (or 1/2) probability of coming up heads. By the product rule, the probability that you will obtain the combined outcome 2 and heads is: (D 2 ) x (P H ) = (1/6) x (1/2) or 1/12 ( [link] ). Notice the word “and” in the description of the probability. The “and” is a signal to apply the product rule. For example, consider how the product rule is applied to the dihybrid cross: the probability of having both dominant traits in the F 2 progeny is the product of the probabilities of having the dominant trait for each characteristic, as shown here:

3 4   ×   3 4   =   9 16

On the other hand, the sum rule    of probability is applied when considering two mutually exclusive outcomes that can come about by more than one pathway. The sum rule states that the probability of the occurrence of one event or the other event, of two mutually exclusive events, is the sum of their individual probabilities. Notice the word “or” in the description of the probability. The “or” indicates that you should apply the sum rule. In this case, let’s imagine you are flipping a penny (P) and a quarter (Q). What is the probability of one coin coming up heads and one coin coming up tails? This outcome can be achieved by two cases: the penny may be heads (P H ) and the quarter may be tails (Q T ), or the quarter may be heads (Q H ) and the penny may be tails (P T ). Either case fulfills the outcome. By the sum rule, we calculate the probability of obtaining one head and one tail as [(P H ) × (Q T )] + [(Q H ) × (P T )] = [(1/2) × (1/2)]+ [(1/2) × (1/2)] = 1/2 ( [link] ). You should also notice that we used the product rule to calculate the probability of P H and Q T , and also the probability of P T and Q H , before we summed them. Again, the sum rule can be applied to show the probability of having just one dominant trait in the F 2 generation of a dihybrid cross:

3 16   +   3 4   =   15 16
The Product Rule and Sum Rule
Product Rule Sum Rule
For independent events A and B, the probability (P) of them both occurring (A and B) is (P A × P B ) For mutually exclusive events A and B, the probability (P) that at least one occurs (A or B) is (P A + P B )

To use probability laws in practice, it is necessary to work with large sample sizes because small sample sizes are prone to deviations caused by chance. The large quantities of pea plants that Mendel examined allowed him calculate the probabilities of the traits appearing in his F 2 generation. As you will learn, this discovery meant that when parental traits were known, the offspring’s traits could be predicted accurately even before fertilization.

Section summary

Working with garden pea plants, Mendel found that crosses between parents that differed by one trait produced F 1 offspring that all expressed the traits of one parent. Observable traits are referred to as dominant, and non-expressed traits are described as recessive. When the offspring in Mendel’s experiment were self-crossed, the F 2 offspring exhibited the dominant trait or the recessive trait in a 3:1 ratio, confirming that the recessive trait had been transmitted faithfully from the original P 0 parent. Reciprocal crosses generated identical F 1 and F 2 offspring ratios. By examining sample sizes, Mendel showed that his crosses behaved reproducibly according to the laws of probability, and that the traits were inherited as independent events.

Two rules in probability can be used to find the expected proportions of offspring of different traits from different crosses. To find the probability of two or more independent events occurring together, apply the product rule and multiply the probabilities of the individual events. The use of the word “and” suggests the appropriate application of the product rule. To find the probability of two or more events occurring in combination, apply the sum rule and add their individual probabilities together. The use of the word “or” suggests the appropriate application of the sum rule.

Questions & Answers

give me somes of examples.
Manu Reply
give example of what.
Ante
example of what
Elizabeth
what are chromosomes?
Ante
what is Biology
Darkoa
Biology is the study of animals and plants on their characters and death
Okol
Ecology is the study of
Abdul Reply
Ecology is the study of the interaction of organism in each other and their environment.
Ante
Ecology is the study of the interaction of organism in each other and their environment
Everlove
oh yes!!! exactly
Great
thank much
Ante
what is biome?
Ante
a geographical area with dinstic species
Francis
thanks much it very helpful
Ante
my pleasure
Francis
characteristics of living things
Jane Reply
excretion, movement, sensitivity, reproduction,growth, respiration,nutrition
Estel
what are there function
Ante
what are their function
Ante
what is the two type of reproduction?
Ante
sexual reproduction and asexual reproduction
Abdul
what are their function
Manu
Hi
Olufunmilokun
What is light independent reaction
Benjamin Reply
correct word or phrase. ______A____ uses carbon dioxide and _____B______ to make sugar during______C______. _____D______ traps light energy during this process. *
Andrea Reply
A_plant ,B_water,C_photosynthesis,D_chlorophyll
Abdul
Plant,sunlight,photosynthesis
Olufunmilokun
by which of the following feature c3 and c4 plant similar A .enzyme used to fix carbon dioxide B.cell where calvin cycle takes place C.cell where carbon dioxide fixation takes place
Kaleab Reply
correct word or phrase. ______A____ uses carbon dioxide and _____B______ to make sugar during______C______. _____D______ traps light energy during this process. *
Andrea
what is biology ?
Bizone Reply
study of organisms
Aurelia
what is link
Lamina Reply
is carbon an organism that matter?
DKP Reply
what is buttress root?
Dora Reply
Buttress root are wide large root
Iyabo
Buttress root/roots are thick roots that emerge out form the base of a large canopy
Musoke
From the base of a large canopy
Musoke
What is biology
Jiboh Reply
Biology is the study of structure and growth of living things and organisms within their biography
Musoke
compare the mechanism of gaseous exchange in an insect and mammal
World Reply
what are the characteristic of livingthing
Joy Reply
Movement Respiration Nutrition Irritability or sensitivity Growth Excretion Reproduction Adaptation Competition Death or Life Span
Emmanuel
Respiration Irritability Movement Excretion Nutrition Growth Reproduction
Amponsah
movement respiration nutrition irritability growth excretion reproduction death
Elizabeth
what is the function of medulla oblongata
mbalenhle Reply
what is the strongest bone in the human body
Chionye
what is biology
Dauda Reply
is a branch of science which deals with the study of living thing
sheka
thanks
Dauda
Biology is got from two main words: bios: means life and Logos: means knowledge therefore, biology is a branch of science that deals with knowledge, life and functioning of living things.
Musoke
what are the characteristics of organism
Seldam
Responds to stimuli
Musoke
Biology is an aspect of science that deals with the. study of living organisms
Amponsah

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Source:  OpenStax, Biology. OpenStax CNX. Feb 29, 2016 Download for free at http://cnx.org/content/col11448/1.10
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