Two-way tables and conditional probability (Statistics) · lesson

Probabilities from the whole table

💡 For a chance out of everyone, count the people who fit and divide by the grand total.

First fill in any missing counts. If 40 are in band and 25 of them are juniors, then 15 band members are not juniors.

P(A) is the count for A over the grand total. P(A and B) counts people in both groups. P(A or B) counts people in either group, each person only once.

🧠 Let's work it out

Of 90 students, 40 are in band, and 25 of those are juniors. Of the 50 not in band, 20 are juniors. Find P(band or junior).

All band members: 40.
Add juniors not in band: 20. That's 60.
60/90 = 2/3.

What do you think comes next?

two-way table: a table that sorts data by two questions at once
probability: the chance something happens, from 0 to 1

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Conditional probability from a table

💡 "Given" shrinks the group: divide by the count of the given group, not the grand total.

P(A | B) is read "the probability of A given B". You already know B happened, so only look at the B group.

The order matters. P(junior | band) looks at band members. P(band | junior) looks at juniors. They usually differ.

🧠 Let's work it out

Use the band table: 40 in band, 25 of them juniors, and 45 juniors in all. Find P(junior | band) and P(band | junior).

P(junior | band) looks only at the 40 band members: 25/40 = 5/8.
P(band | junior) looks only at the 45 juniors: 25/45 = 5/9.

What do you think comes next?

conditional probability: the chance of one event when you know another happened
given: already known to be true; written with a bar, |

The conditional probability formula

💡 P(A | B) = P(A and B) ÷ P(B), and P(A and B) = P(A) × P(B | A).

Without a table, use the formula. Divide the chance of both by the chance of the given event.

If you know P(B | A) instead, first multiply to get P(A and B). Then divide by P(B).

🧠 Let's work it out

P(A) = 0.6, P(B | A) = 0.5, and P(B) = 0.5. Find P(A | B).

P(A and B) = 0.6 × 0.5 = 0.3.
P(A | B) = 0.3 ÷ 0.5.
That is 0.6.

What do you think comes next?

P(A and B): the chance that both events happen

Testing independence

💡 Two events are independent when knowing one doesn't change the chance of the other: P(A | B) = P(A).

Compare a conditional probability with the overall one. If they match, the events are independent. If not, they are connected.

Don't compare raw counts. Groups of different sizes can make counts look different even when the rates are the same.

🧠 Let's work it out

Are being in band and being a junior independent?

P(junior | band) = 25/40 = 0.625.
P(junior) = 45/90 = 0.5.
0.625 is not 0.5, so knowing someone is in band changes the chance.
So they are not independent.

What do you think comes next?

independent: not affecting each other's chances