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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

where is the pectoral gridle located?
Tiania Reply
What is hypotonic
Bright Reply
what is hypotonic
Dangaya
the difference between the two cells
Obeng Reply
explain the courses and the correction of lon term sightedness and short term sightedness
Isaac Reply
why drinking excess alcohol causes thirst and dehydration
uwikuzo Reply
Can we chat about nutrition please?
Elia
yes
Uzair
sure
Uzair
Uhm why is it so important to follow the nutritional process?
Elia
what is reproduction
smart Reply
it is d act of bringing young ones to life
Oyebanji
to ensure survival of a species🚴‍♀️
Michelle
what is a genotype
Collins
what is hazardous
smart
a cell is the smallest unit of a living thing. so we all have cell
smart
It is the formation of a zygote resulting from the fusion of the sperm cell with the ovum.Thus,this results in the production of new species which are genetically dissimilar from their parent cells.
Pallavi
what is size of cell
Mohd Reply
what is size of Hart
Mohd
nanometers=um sign thingie
Michelle
microns=nanometers
Michelle
monomers and polymers of nucleic acids?
Jyrl Reply
dna and rna involvement
Michelle
give me the elements of the soil
Iguma Reply
Air, water, organic matter, inorganic matter
Tshwano
soil water humus air
Veronicah
yap
Beloved
silica, iron
Patience
potassium, sulfur, calcium, carbon
Michelle
what is cell
iyaji Reply
A cell is a smallest fundamental unit of a living organisms.
Kem
the basic structural and functional unit of life
Patience
what is size of cell
Mohd
all things are made up of.....all things cannot exist without pre-exisiting cells...check out the 16th ce tury to learn more about microscope use and cells. i will give a hint: Mr. L.
Michelle
we are all made of cells
Michelle
Nutrition - sensitive intervention
Therowda Reply
what does the sori In fern mean
arhin Reply
biology is the study .exactly what is life?
Jerry
i'm sorry , the study of life
Jerry
How is food distributed to all parts of cytoplasm
Therowda
how can U identify a person through his blood
Frankyx Reply
through genetic fingerprinting where specific DNA sequences in a person genome can be identified.
Pallavi
by help of genetics and DNA test
Meerbadini
can it also be detected using an RNA test
Frankyx
environmental biology
Ojesola Reply
what is phytoplankton
Ojesola
What is anabolism
Treasure Reply
the break down of substance
Aunty

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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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