Quick Notes

bays therom

Hypothesis = H
Evidence = E

Three ways of saying the same thing. “Not E”

E prime == E’
E complement == Ec
Not E == ¬E

Example Problem 1:

Bob and Jane are sometimes late for work. 70% of the time neither of them are late. Bob is late 20% of the time, while Jane is late 25% of the time.

Last Monday Jane was late. Find the probability that Bob was late.

Evidence: Jane is late

Hypothesis: Bob is late

P(E' ⋂ H') = 0.7 => (70% of the time neither of them are late)

P(H) = 0.2 => (Bob is late 20% of the time)

P(E) = 0.25 => (Jane is late 25% of the time)

Step 1: Find out how much of the total both H and (E' ⋂ H')

0.2 + 0.7 = 0.9

Step 2: How much of E that is not part of H is there?

1 - 0.9 = 0.1 i.e. (E ⋂ H') = 0.1

Step 3: find what part of E intersects with H 0.25 - 0.1 = 0.15

Step 4: find (E' ⋂ H) 0.2 - 0.15 = 0.05

Step 5: find the probability that Bob is late given the evidence that Jane is late

Formula: P(H ⋂ E) / P(E)

0.15 / 0.25 = 3 / 5 = 60%

We now know that Bob has a 60% chance of being late when Jane is late. When Jane is not late he has a 20% chance of being late.

Here is a Venn Diagram to help visualize

bays therom

Example Problem 2:

Bob loves to play Tennis, but especially when the weather is good. When it is sunny, the probability that he plays tennis is 80%. When it is not sunny, the probability is just 35%. There is a 60% chance that it is sunny on any given day. Last Saturday he played tennis. What is the probability that it was sunny last Saturday?

H: It was sunny last Saturday.
E: Bob played tennis last Saturday.

Create a tree with all possibilities and their complement percentages. bays tree

Probability that it was sunny and Bob played tennis can be seen by following the right side of

the tree: P(H ⋂ E) = 0.6 X 0.8 = 0.48

To calculate E by itself without H we can say:

Either it was sunny and Bob played tennis (H ⋂ E)

Or it may not have been sunny and Bob played tennis (H’ ⋂ E).

We then add these two things together (H ⋂ E) + (H’ ⋂ E)

(H ⋂ E) = 0.6 X 0.8 = 0.48

(H’ ⋂ E) = 0.4 X 0.35 = 0.14

P(E) = 0.48 + 0.14 = 0.62

apply Bayes theorem to get: H = 0.48/0.62 = ~77%

We now know it is 77% likely that It was sunny last Saturday

Example Problem 3:

During recessions there is a 40% chance that Bob will lose his job. Otherwise, there is a 5% chance.

In any given year there is a 10% chance of a recession.

Last year Bob lost his job. Find the probability that there was a recession.

E = Last year Bob lost his job
H = There was a recession

Build a tree with all possibilities.

bays tree

Find the probability that he lost his job and there is a recession (H ⋂ E)

bays tree

The probability of E is either there was a recession and he lost his job (H ⋂ E) Or there was not a recession and he lost his job (H’ ⋂ E)

bays tree

= ~47%