Lesson 5 — Probability


Probability

Probabilities usually are expressed as real numbers with values from zero to one. They may be expressed as fractions or percentages, too. A probability of zero means no chance; a probability of one means it is a sure thing. Anywhere between, the chance will differ: low values indicate it is unlikely, and high values indicate it is likely.

The two general rules when considering probability are the rule of addition and the rule of product. Deciding when to use which rule depends on whether the probabilities are linked or independent.



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Addition Rule (OR)

If two events are mutually exclusive, the probability that either will occur is their sum. Two events are mutually exclusive when either event can occur but they cannot occur at the same time.


Example

What is the probability of rolling a 3 or a 4 on a single die? Rolling a 3 and rolling a 4 on a single die are mutually exclusive events. You can either roll a 3 or a 4, but you cannot roll both numbers on a single roll with one die.

  • The probability of rolling a 3 on a single die is «math»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«/math». A die has six sides; therefore, there are six possibilities.

  • The probability of rolling a 4 is the same, «math»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«/math».

  • Thus, the probability of rolling either a 3 or a 4 is «math»«mfrac»«mn»2«/mn»«mn»6«/mn»«/mfrac»«mo»§#160;«/mo»«mi»o«/mi»«mi»r«/mi»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«/math».       «math»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mo»§#160;«/mo»«mo»+«/mo»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mn»2«/mn»«mn»6«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«/math»


Having many exclusive outcomes increases the probability.


Product Rule (AND)

If two outcomes are independent of each other, the probability that both will occur is their product. Two events are independent of each other if one event does not affect the outcome of the second event.


Example

What is the probability of rolling two 6s in a row on a single die?

  • The probability of rolling a 6 on a single die is «math»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«/math», and the probability of rolling another 6 is the same. These two events are independent because rolling a 6 does not affect the probability of rolling another 6.

  • The probability of rolling two 6s is «math»«mfrac»«mn»1«/mn»«mn»36«/mn»«/mfrac»«/math».        «math»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mo»§#160;«/mo»«mo»§#215;«/mo»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»36«/mn»«/mfrac»«/math»


Having many independent outcomes occur at once decreases the likelihood, or probability.


Practice Problem

What is the probability of obtaining "Heads - Tails - Tails" in a coin toss?

The product rule will be applied. Take the probability of the first event, multiply by the second, and multiply by the third.

A coin toss has two possibilities: heads or tails. Therefore, the probability of tossing heads is «math»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/math» and the probability of tossing tails is «math» «mfrac» «mn»1«/mn» «mn»2«/mn» «/mfrac» «/math».

The probability of tossing heads - tails - tails is «math»«mfrac»«mn»1«/mn»«mn»8«/mn»«/mfrac»«/math». (The probability of three tosses of any other combination is the same. )

«math»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»§#160;«/mo»«mo»§#215;«/mo»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»§#160;«/mo»«mo»§#215;«/mo»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»8«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»0«/mn»«mo».«/mo»«mn»125«/mn»«mo»§#160;«/mo»«mi»p«/mi»«mi»r«/mi»«mi»o«/mi»«mi»b«/mi»«mi»a«/mi»«mi»b«/mi»«mi»i«/mi»«mi»l«/mi»«mi»i«/mi»«mi»t«/mi»«mi»y«/mi»«mo»§#160;«/mo»«mi»o«/mi»«mi»r«/mi»«mo»§#160;«/mo»«mn»12«/mn»«mo».«/mo»«mn»5«/mn»«mo»%«/mo»«mo»§#160;«/mo»«mi»c«/mi»«mi»h«/mi»«mi»a«/mi»«mi»n«/mi»«mi»c«/mi»«mi»e«/mi»«/math»


Watch and Listen

Watch the following video on probability.



 ©Alberta Education. Classical Genetics and Dihybrid Crosses: The Choreography of Genetic Inheritance (19:25-28:50); Series 26.  LearnAlberta.ca

 


  1. What is the probability of rolling a 3on one die?

  2. What does the sum of all possibilities equal?

  3. What is the probability of rolling a 6at the same time on each of two dice? To determine this, what "rule" must you use?

  4. How can you use this rule and two smaller Punnett squares to predict the offspring of a dihybrid cross?

  5. If you assume that the parents in the film are heterozygous for all three traits, what is the probability of having a child with widow's peak, hitchhiker's thumb, and free earlobes?

  1. The probability of rolling a 3 is 1 in 6 or 0.167.

  2. The sum of all possibilities always equals 1.

  3. By using the Product Rule, we can determine the probability of rolling a 6 on two different dice. The probability of rolling a 6 on one die is «math»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«/math». The probability of rolling a 6 on both dice is «math»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mo»§#160;«/mo»«mo»§#215;«/mo»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»36«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»0«/mn»«mo».«/mo»«mn»028«/mn»«/math». 

  4. You can use the probability of a monohybrid Punnett square and multiply it with a second probability of a monohybrid Punnett square to obtain the same result as a dihybrid Punnett square.

  5. Widow's peak is dominant, hitchhiker's thumb is recessive, and free earlobe is the dominant allele. In a monohybrid Punnett square of two heterozygous individuals, 3 of 4 offspring will be dominant and 1 of 4 will be recessive. Using the Product Rule,  the probability of having a child  with widow's peak, hitchhiker's thumb, and free earlobes is «math»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«mo»§#160;«/mo»«mo»§#215;«/mo»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mo»§#160;«/mo»«mo»§#215;«/mo»«mo»§#160;«/mo»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mn»9«/mn»«mn»64«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»0«/mn»«mo».«/mo»«mn»14«/mn»«/math».