Probability

• Can you give some examples of the importance of the probability concept in biology ?

• Sex of babies in humans/animals (heterogametic sex)

• DNA mutation rate (hypermutator)

• Probability is essential because we use samples to investigate the world

• As we have seen, chance plays a major role in the properties of samples

• e.g., 95% confidence interval of the mean, Proba to capture/see a bird given that it is present (estimating abundance), etc.

• Here we will discuss basic probability calculations

Definition: A random trial is a process or experiment that has two or more possible outcomes whose occurrence cannot be predicted with certainty.

Definition: An event is any potential subset of all the possible outcomes of a random trial.

Definition: The probability of an event is the proportion of times the event would occur if we repeated a random trial over and over again under the same conditions. Probability ranges between zero and one.

Definition: The probability of an event not occurring is one minus the probability that it occurs. \[ \mathrm{Pr[{\it not}\ A]} = 1-\mbox{Pr[A]} \]

Definition: General addition rule \[ \mathrm{Pr[A \ or \ B]} = \mathrm{Pr[A]} + \mathrm{Pr[B]} - \mathrm{Pr[A \ and \ B]} \]

Exemple: blood type 0 or Rhesus factor + \[ \mathrm{Pr[O \ or \ +]} = \mathrm{Pr[O]} + \mathrm{Pr[+]} - \mathrm{Pr[O \ and \ +]} \]

Definition: The conditional probability of an event is the probability of that event occurring given that another event has already occurred.

Definition: The conditional probability of an event B given that A occurred is \[ \mathrm{Pr[B \ | \ A]} = \frac{\mathrm{Pr[A \ and \ B]}}{\mathrm{Pr[A]}} \]

Definition: General multiplication rule \[ \mathrm{Pr[A \ and \ B]} = \mathrm{Pr[A]}\times\mathrm{Pr[B \ | \ A]} \]

Commonly confused!

Definition: Two events are mutually exclusive if they cannot both occur at the same time. \[ \mathrm{Pr[A \ and \ B]} = 0 \]

Definition: Two events are independent if the occurrence of one does not inform us about the probability that the second will occur. \[ \mathrm{Pr[B \ | \ A]} = \mathrm{Pr[B]} \]

These two conditions simplify the general addition and multiplicative rules:

If two events are mutually exclusive, then \[ \mathrm{Pr[A \ or \ B]} = \mathrm{Pr[A]} + \mathrm{Pr[B]} \]

Exemple: blood type 0

\[ \mathrm{Pr[O]} = \mathrm{Pr[O+ \ or \ O-]} = \mathrm{Pr[O+]} + \mathrm{Pr[O-]} - \mathrm{Pr[O+ \ and \ O-]} \]

\[ \mathrm{Pr[O]} = \mathrm{Pr[O+]} + \mathrm{Pr[O-]} \]

If two events are independent, then \[ \mathrm{Pr[A \ and \ B]} = \mathrm{Pr[A]} \times \mathrm{Pr[B]} \]

\[ \mathrm{Pr[2 \ six]} = \mathrm{Pr[first \ roll \ is \ six \ and \ 2nd \ roll \ is \ six]} = 1 / 6 \times 1 / 6 = 1 / 36 \]

• Probability trees

• Law of total probability

• Contingency tables

• Probability distributions

\[ \mathrm{Pr[P \ and \ M]} = \mathrm{Pr[P]}\times\mathrm{Pr[M \ | \ P]} = 0.18 \]

Definition: The law of total probability is given by \[ \begin{align*} \mathrm{Pr[A]} & = \sum_{All \ values \ of \ B}\mathrm{Pr[A \ and \ B]} \\ & = \sum_{All \ values \ of \ B} \mathrm{Pr[B]}\ \mathrm{Pr[A\ | \ B]}, \end{align*} \] where \( B \) represents all possible mutually exclusive values of the condition

So What is \( \mathrm{Pr[M]} \)?

Definition: The law of total probability is given by \[ \mathrm{Pr[A]} = \sum_{All \ values \ of \ B} \mathrm{Pr[B]}\ \mathrm{Pr[A\ | \ B]} \]

\( \mathrm{Pr[M]} = \mathrm{Pr[P]}\times\mathrm{Pr[M \ | \ P]} + \mathrm{Pr[NP]}\times\mathrm{Pr[M \ | \ NP]} \)

\( \mathrm{Pr[M]} = (0.20\times 0.90) + (0.80\times 0.05) = 0.22 \)

\( (\mathrm{Pr[P \ and \ M]}=0.18)\neq (\mathrm{Pr[P]}\times\mathrm{Pr[M]}= 0.20\times 0.22 = 0.044) \)

The probability to get a male depends on whether the host was already parasitized, i.e. the events are not independent

Smoking and cancer contingency table

            health
status       cancer not cancer    Sum
  smoker       8944      43056  52000
  not smoker    624      47376  48000
  Sum          9568      90432 100000

Question: What is Pr[smoker]?

Answer: 52000/100000 = 0.52

Question: What is Pr[cancer]?

Answer: 9568/100000 = 0.09568

Smoking and cancer contingency table

            health
status       cancer not cancer    Sum
  smoker       8944      43056  52000
  not smoker    624      47376  48000
  Sum          9568      90432 100000

Question: What is Pr[cancer | smoker]?

Answer: 8944/52000 = 0.172

Question: What is Pr[smoker | cancer]?

Answer: 8944/9568 = 0.9347826

Smoking and cancer contingency table

            health
status       cancer not cancer    Sum
  smoker       8944      43056  52000
  not smoker    624      47376  48000
  Sum          9568      90432 100000

Question: What is Pr[smoker AND cancer]?

Answer: 8944/100000 = 0.08944

Definition: A probability distribution is a list of the probabilities of all mutually exclusive outcomes of a random trial.

Unlike discrete probability distributions, the height of a continuous probability curve (say, at Y = 1) is not the probability of obtaining Y = 1.

Instead, the probability to obtain a value of Y within some range is given by the area under the curve.

Probability densities