M1.4 Understand simple probability

The difference between probability and chance 

In statistics we use probability as the appropriate term for expressing likelihoods (as ratios, decimals or percentages).

For example – When tossing a coin we have a probability of 0.5 of the coin landing on heads – with this information we can predict that with ten coin flips, approximately five of them will land heads up. 

In everyday speech ‘chance’ is often used to mean the same thing as ‘probability’. For example we might say ‘I think the chance of rain today is about 50/50’ meaning that we think there is a 50% probability of rain. However, when we use the word ‘chance’ in commenting on statistics we are using it to talk about the random deviations from probability that can occur (and especially whether we think these random deviations are sufficient to explain why the outcome did not exactly match our expectation based on probability). 

Using coin tosses as an example once more, if we actually got six heads from ten tosses of the coin we might say ‘getting six rather than the predicted five was due to chance’. If we got nine heads we might say ‘this is too far from the predicted outcome to be explained by chance and so we now believe this coin is biased’. The cut-off point for how far away from the prediction (based on probability) the results have to be before they cannot be explained by chance is where statistical tests come in (see M1.9). 

The effect of chance, in percentage terms, is largest when the number of repeats is small. Clearly if we only toss a coin four times, a single result has a big effect on the percentage of heads whereas if we do 1000 repeats the effect of a single result is much smaller. This is the reason that experiments often use many repeats, or take many samples. 

Probability and patterns of genetic inheritance in a monogenetic cross.

In a genetic cross of two Drosophila melanogaster (fruit flies) that are each heterozygous for the mutant allele of the vestigial gene (which is recessive and found on chromosome 2), we predict that the probability of each individual offspring being a recessive homozygote (which means it will have tiny wings) is 0.25, the equivalent of saying that 1 in 4 offspring will have tiny wings. 

Probability and patterns of genetic inheritance in a dihybrid cross.

Things get a little more complicated when we include a second genetic trait. 

Individuals that are homozygous for the recessive mutant version of the ebony gene (found on chromosome 3), have very darkly coloured bodies (Heterozygotes sometimes have slightly darker bodies). If the parents are heterozygous at both the ebony locus and the vestigial locus, then the probability of an individual having tiny wings and dark bodies is 0.25 x 0.25 = 0.0625 or 1/16. 

Again this can be shown in a Punnett square giving the 16 equally likely outcomes: 

And the numbers of each of those outcomes giving each of the four possible phenotypes is easier to see if we colour code: 

So from our probability calculation, illustrated with the Punnett square we can say, 

for each offspring there is a 1/16 or 0.0625 or 6.25% probability that it will be doubly homozygous for the recessive alleles and therefore display the double mutant phenotype of tiny wings and an ebony-coloured body. 

Similarly there is a 3/16 probability of each individual having normal wings but an ebony-coloured body. 

And 3/16 probability of each individual having tiny wings but a normal-coloured body. 

There is a 9/16 probability of each offspring appearing normal (although of course many of these flies will be heterozygotes, carrying mutant alleles at one or both loci). 

This leads to the classic expected ratio of phenotypes in the offspring in this kind of breeding experiment:  

9 : 3 : 3 : 1 

What if the results you get from such an experiment don’t exactly match the ratio you’d expect based on your probability calculations? 

It could just be due to chance or it could indicate that your assumptions were wrong. Statistical tests (in this case the chi-squared test) (see M1.9) will help you decide if the deviation away from your expectation is big enough to demand a re-think of your assumptions. 

If so, maybe the loci are linked (on the same chromosome) so they won’t be inherited truly independently. Or maybe something else weird is happening (e.g. the double mutant offspring are not surviving to adulthood to be counted).  

M1.4 – Understand simple probability 

Students Quiz 

1. What is the probability of rolling a 5 on a six-sided die?  

2. What is the probability of rolling a 3 or a 5 on a six-sided die?  

3. What is the probability of rolling at least one 3 when rolling two dice?  

4.  What is the probability of rolling two 3s one after the other when rolling a single die?  

5. We have two cats where one is homozygous for alleles that produce a short tail which is a recessive trait, while the other is homozygous for alleles that produce a long tail which is dominant. With this knowledge we can make predictions about the characteristics of any offspring produced when these two cats are bred together.  

1. What is the probability that offspring would inherit one copy of the short tail allele?  

2. What is the probability that offspring would have short tails?  

6.  I have two flowers, both heterozygous at two loci. One locus is the petal colour gene where the yellow allele is recessive and the red allele is dominant.   The other locus is the petal shape gene where a smooth petal allele is dominant and a wrinkly petal allele is recessive. 

A. What is the probability of an individual inheriting just one copy of the wrinkly allele and just one copy of the red allele? (i.e. being doubly heterozygous just like the parents) 

B. What is the probability of seeing an individual from this cross which has wrinkly red petals?  

Answers to quiz