Computation of conditional probabilitiesMultiplication theorem

N.I. Lobachevsky State Universityof Nizhni Novgorod

Probability theory and mathematical statistics:

Conditional probability — Practice

Associate ProfessorA.V. Zorine

Conditional probability — Practice 1 / 11

Computation of conditional probabilitiesMultiplication theorem

Two dice are rolled. What’s the conditional probability that both diceare ’s if it’s known that the sum of points is divisible by 5?

Solution 1. A ∩ B = (5, 5),B = (1, 4), (2, 3), (3, 2), (4, 1), (4, 6), (5, 5), (6, 4),

P(A|B) =17

Solution 2. Ω = (1, 1), (1, 2), . . . , (6, 6),

P(A ∩ B) =1

36, P(B) =

736

,

P(A|B) =P(A ∩ B)

P(B)=

17

Conditional probability — Practice 2 / 11

Computation of conditional probabilitiesMultiplication theorem

Two dice are rolled. What’s the conditional probability that both diceare ’s if it’s known that the sum of points is divisible by 5?

Solution 1. A ∩ B = (5, 5),B = (1, 4), (2, 3), (3, 2), (4, 1), (4, 6), (5, 5), (6, 4),

P(A|B) =17

Solution 2. Ω = (1, 1), (1, 2), . . . , (6, 6),

P(A ∩ B) =1

36, P(B) =

736

,

P(A|B) =P(A ∩ B)

P(B)=

17

Conditional probability — Practice 2 / 11

Computation of conditional probabilitiesMultiplication theorem

Two dice are rolled. What’s the conditional probability that both diceare ’s if it’s known that the sum of points is divisible by 5?

Solution 1. A ∩ B = (5, 5),B = (1, 4), (2, 3), (3, 2), (4, 1), (4, 6), (5, 5), (6, 4),

P(A|B) =17

Solution 2. Ω = (1, 1), (1, 2), . . . , (6, 6),

P(A ∩ B) =136

, P(B) =736

,

P(A|B) =P(A ∩ B)

P(B)=

17

Conditional probability — Practice 2 / 11

Computation of conditional probabilitiesMultiplication theorem

An urn has 5 red balls and 8 blue balls. Two balls are taken out, Letevent A occur when the first ball is red and B when the second ball isred. Compute P(B|A).

After A occurred, there are only 4 red balls and only 12 balls in total.So

P(B|A) =412

Conditional probability — Practice 3 / 11

Computation of conditional probabilitiesMultiplication theorem

An urn has 5 red balls and 8 blue balls. Two balls are taken out, Letevent A occur when the first ball is red and B when the second ball isred. Compute P(B|A).

After A occurred, there are only 4 red balls and only 12 balls in total.So

P(B|A) =412

Conditional probability — Practice 3 / 11

Computation of conditional probabilitiesMultiplication theorem

Let Ω = ω1, ω2, ω3, A = ω1, ω3, B = ω2, ω3, p(ω1) = 0,2,p(ω2) = 0,2, p(ω3) = 0,6. Calculate P(A|B).

A ∩ B = ω3, P(A ∩ B) = P(ω3) = p(ω3) = 0,6,P(B) = p(ω2) + p(ω3) = 0,8,

P(A|B) =0,60,8

= 0,75

Conditional probability — Practice 4 / 11

Computation of conditional probabilitiesMultiplication theorem

Let Ω = ω1, ω2, ω3, A = ω1, ω3, B = ω2, ω3, p(ω1) = 0,2,p(ω2) = 0,2, p(ω3) = 0,6. Calculate P(A|B).

A ∩ B = ω3, P(A ∩ B) = P(ω3) = p(ω3) = 0,6,P(B) = p(ω2) + p(ω3) = 0,8,

P(A|B) =0,60,8

= 0,75

Conditional probability — Practice 4 / 11

Computation of conditional probabilitiesMultiplication theorem

One card is take from a pile of 52 cards. Let event A occur when anace is taken out, event B when a card of spades is taken out. ComputeP(A|B) and P(B|A).

P(A) =452

, P(B) =1252

, P(A ∩ B) =152

,

P(A|B) =1

521252

, P(B|A) =152452

Conditional probability — Practice 5 / 11

Computation of conditional probabilitiesMultiplication theorem

One card is take from a pile of 52 cards. Let event A occur when anace is taken out, event B when a card of spades is taken out. ComputeP(A|B) and P(B|A).

P(A) =452

, P(B) =1252

, P(A ∩ B) =152

,

P(A|B) =1

521252

, P(B|A) =152452

Conditional probability — Practice 5 / 11

Computation of conditional probabilitiesMultiplication theorem

Three random numbers are chosen without replacement from the set1, 2, . . . , N. What’s the conditional probability that the thirdnumber is between the first and the second one if it is known that thefirst number is smaller then the second one.

Ω = (x1, x2, x3) : x1 6= x2, x1 6= x3, x2 6= x3, x1, x2, x3 ∈1, 2, . . . , N, A = x1 < x3 < x2 or x2 < x3 < x1, B = x1 < x2.N(B) = C2

N(N − 2), N(A ∩ B) = C3N ,

P(A|B) =C3

N

C2N(N − 2)

=13

Conditional probability — Practice 6 / 11

Computation of conditional probabilitiesMultiplication theorem

Three random numbers are chosen without replacement from the set1, 2, . . . , N. What’s the conditional probability that the thirdnumber is between the first and the second one if it is known that thefirst number is smaller then the second one.

Ω = (x1, x2, x3) : x1 6= x2, x1 6= x3, x2 6= x3, x1, x2, x3 ∈1, 2, . . . , N, A = x1 < x3 < x2 or x2 < x3 < x1, B = x1 < x2.N(B) = C2

N(N − 2), N(A ∩ B) = C3N ,

P(A|B) =C3

N

C2N(N − 2)

=13

Conditional probability — Practice 6 / 11

Computation of conditional probabilitiesMultiplication theorem

There is a set of 100 cards with numbers 00, 01, . . . , 99. A card istaken at random from the set. Let h1 and h2 denote the sum and theproduct of the digits of a chosen card. Compute P(h1 = i|h2 = 0),i = 0, 2, . . . , 18.

h2 = 0 = 00, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 20, 30,

40, 50, 60, 70, 80, 90

h1 = 0 and h2 = 0 = 00, P(h1 = 0|h2 = 0) =119

,

h1 = 1 and h2 = 0 = 01, 10, P(h1 = 1|h2 = 0) =2

19,

. . .

h1 = 9 and h2 = 0 = 09, 90, P(h1 = 1|h2 = 0) =2

19,

h1 = 11 and h2 = 0 = = ∅, P(h1 = 1|h2 = 0) = 0,

. . .

Conditional probability — Practice 7 / 11

Computation of conditional probabilitiesMultiplication theorem

There is a set of 100 cards with numbers 00, 01, . . . , 99. A card istaken at random from the set. Let h1 and h2 denote the sum and theproduct of the digits of a chosen card. Compute P(h1 = i|h2 = 0),i = 0, 2, . . . , 18.

h2 = 0 = 00, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 20, 30,

40, 50, 60, 70, 80, 90

h1 = 0 and h2 = 0 = 00, P(h1 = 0|h2 = 0) =119

,

h1 = 1 and h2 = 0 = 01, 10, P(h1 = 1|h2 = 0) =2

19,

. . .

h1 = 9 and h2 = 0 = 09, 90, P(h1 = 1|h2 = 0) =2

19,

h1 = 11 and h2 = 0 = = ∅, P(h1 = 1|h2 = 0) = 0,

. . .

Conditional probability — Practice 7 / 11

Computation of conditional probabilitiesMultiplication theorem

A random point ω = (x, y) is chosen in a square

G = (x, y) : 0 6 x, y 6 1.

LetA1 =

x 6

12

⊂ G, A2 =

y 6

12

⊂ G,

A3 =(

x− 12

)(y− 1

2

)< 0⊂ G

Compute P(A1|A2), P(A3|A1), P(A3|A1 ∩ A2).

Use geometric probability. Area of G is 1, area of A1 is 12 , area of A2

is 12 , area of A1 ∩ A2 is 1

4 , area of A3 ∩ A1 is 14 , area of

A1 ∩ A2 ∩ A3 = ∅ is 0.

P(A1|A2) =12, P(A3|A1) =

12, P(A3|A1 ∩ A2) = 0.

Conditional probability — Practice 8 / 11

Computation of conditional probabilitiesMultiplication theorem

A random point ω = (x, y) is chosen in a square

G = (x, y) : 0 6 x, y 6 1.

LetA1 =

x 6

12

⊂ G, A2 =

y 6

12

⊂ G,

A3 =(

x− 12

)(y− 1

2

)< 0⊂ G

Compute P(A1|A2), P(A3|A1), P(A3|A1 ∩ A2).

Use geometric probability. Area of G is 1, area of A1 is 12 , area of A2

is 12 , area of A1 ∩ A2 is 1

4 , area of A3 ∩ A1 is 14 , area of

A1 ∩ A2 ∩ A3 = ∅ is 0.

P(A1|A2) =12, P(A3|A1) =

12, P(A3|A1 ∩ A2) = 0.

Conditional probability — Practice 8 / 11

Computation of conditional probabilitiesMultiplication theorem

There are 5 ”good” questions among 25 prepared for a test (a questionis good if a student knows the answer). Two students take onequestion in turn. What’s the probability that a) the first student took a”good” question, b) both students have ”good” questions?

a)525

b)5

25· 4

24

Conditional probability — Practice 9 / 11

Computation of conditional probabilitiesMultiplication theorem

There are 5 ”good” questions among 25 prepared for a test (a questionis good if a student knows the answer). Two students take onequestion in turn. What’s the probability that a) the first student took a”good” question, b) both students have ”good” questions?

a)525

b)5

25· 4

24

Conditional probability — Practice 9 / 11

Computation of conditional probabilitiesMultiplication theorem

Two players take balls in turn from an urn with M red and N −Mbrown balls in it. A player wins if he gets a red ball. What’s theprobability the first player wins? Consider N = 4, M = 1.

Let event Ak occur when the first player takes a red ball at the k-thturn, Bk occur when the second player takes a red ball at the k-th turn,A occur if the first player wins.

A = A1 ∪ A1 ∩ B1 ∩ A2 ∪ A1 ∩ B1 ∩ A2 ∩ B2 ∩ A3 ∪ . . .

P(A) =MN

+N −M

N· N −M − 1

N − 1· M

N − 2

+N −M

N· N −M − 1

N − 1· N −M − 2

N − 2· N −M − 3

N − 3· M

N − 4+ . . .

Conditional probability — Practice 10 / 11

Computation of conditional probabilitiesMultiplication theorem

Two players take balls in turn from an urn with M red and N −Mbrown balls in it. A player wins if he gets a red ball. What’s theprobability the first player wins? Consider N = 4, M = 1.

Let event Ak occur when the first player takes a red ball at the k-thturn, Bk occur when the second player takes a red ball at the k-th turn,A occur if the first player wins.

A = A1 ∪ A1 ∩ B1 ∩ A2 ∪ A1 ∩ B1 ∩ A2 ∩ B2 ∩ A3 ∪ . . .

P(A) =MN

+N −M

N· N −M − 1

N − 1· M

N − 2

+N −M

N· N −M − 1

N − 1· N −M − 2

N − 2· N −M − 3

N − 3· M

N − 4+ . . .

Conditional probability — Practice 10 / 11

Computation of conditional probabilitiesMultiplication theorem

The probability for a letter to be in the desk’s drawers equals p, and ifit’s in the drawers then each drawer has equal probability to have theletter. There 8 drawers in the desk. What’s the probability to find theletter in the 8-th drawer?

Event A occurs when the letter is in the 8-th drawers, event B occurswhen the letter is in the desk’s drawers.

P(A) = P(A ∩ B) = P(B)P(A|B) = p · 18

Conditional probability — Practice 11 / 11

Computation of conditional probabilitiesMultiplication theorem

The probability for a letter to be in the desk’s drawers equals p, and ifit’s in the drawers then each drawer has equal probability to have theletter. There 8 drawers in the desk. What’s the probability to find theletter in the 8-th drawer?

Event A occurs when the letter is in the 8-th drawers, event B occurswhen the letter is in the desk’s drawers.

P(A) = P(A ∩ B) = P(B)P(A|B) = p · 18

Conditional probability — Practice 11 / 11