The joint chance is the chance of the intersection of two occasions or the chance of two occasions occurring collectively.

Let A and B be two occasions in a pattern house.

The intersection of A and B is the gathering of all outcomes which might be frequent to each A and B.

We will denote the intersection of A and B as  A  B or  AB.

The chance of A and B occurring collectively is P(A  B)

Check out the contingency desk of impartial occasions.

Move = {66, 54}  and Males = {66, 44}

Move  Males = {66}

Now how can we compute the joint chance? Utilizing the 2 tables under, we’ll compute P(cross / male)

Contingency table of dependent events

Here’s what we discovered Within the lesson about chance of impartial occasions.

Utilizing the contingency desk of impartial occasions

P(cross / male) =

66
/
110

Utilizing the contingency desk of dependent occasions

P(cross / male) =

46
/
102

There may be one other technique to discover these solutions.

Did you discover this utilizing the desk of impartial occasions?

$$
frac{frac{66} {200}} {frac{110} {200}} = frac{66} {200} × frac{200} {110} = frac{66} {110} $$

Did you discover this utilizing the desk of dependent occasions?

$$
frac{frac{46} {200}} {frac{102} {200}} = frac{46} {200} × frac{200} {102} = frac{46} {102} $$

In conclusion, 

$$ P(cross / male) =
frac{frac{66} {200}} {frac{110} {200}} $$

In conclusion, 

$$ P(cross / male) =
frac{frac{46} {200}} {frac{102} {200}} $$

Joint probability

P(cross and male) =

66
/
200

P(cross and male) =

46
/
200

As you may see, it doesn’t matter if the occasions are impartial or not, the components is

P(cross / male) =

P(cross and male)
/
P(male)

Multiplication rule of joint occasions

Multiply each side of the equation instantly above by P(male)

P(male) × P(cross / male) =

P(cross and male)
/
P(male)

× P(male)

P(male) × P(cross / male) = P(cross and male)

P(cross and male )  = P(male) × P(cross / male)

Normally, if A and B is the intersection of two occasions.

P(A and B) = P(A) × P(B / A)  or P(A and B) = P(B) × P(A / B)

Joint chance of impartial occasions

If A and B are impartial occasions, we all know that P(A) = P(A / B)  or  P(B) = P(B / A)

P(A and B) = P(A) × P(B / A).

Since  P(B / A) = P(B), P(A and B) = P(A) × P(B)

Joint chance of mutually unique occasions

The joint chance of two mutually unique occasions is all the time zero.

When two occasions A and B are mutually unique, A  B = { }

In different phrases, the intersection is empty. Because the intersection is empty, the chance zero.

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