Mathematics: AI HL
Paper 3
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Mathematics: AI HL
Available time 20 Minute(s)
01

This question asks you to investigate the number of throws of a die up to and including a certain result.

Paper 3
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Available time 20 Minute(s)
01. Question
Level Hard
Maximum Mark 31
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This question asks you to investigate the number of throws of a die up to and including a certain result.

Throw an unbiased 6-sided die until you get the first 6. Let X= number of throws.

(a). (i) Find P(X=1).

[Marks: 1]

(a). (ii) Explain why P(X=4)=56316.

[Marks: 2]

(a). (iii) Find P(X<9).

[Marks: 2]

Let p=56, q=16

(b). (i) Express E(X) as an infinite series with p and q.

[Marks: 2]

(b). (ii) By using the fact that q=1-p, or otherwise, find E(X).

[Marks: 3]

(c). (i) Show thatVar(X)=2p1+2p+3p2++1+p+p2+p3+-36

[Marks: 4]

(c). (ii) By considering ddtt+t2+t3+, or otherwise, find Var(X).

[Marks: 3]

Throw the die until you get the same score two consecutive times. Let Y= number of throws.

(d). (i) Find P(Y=2).

[Marks: 2]

(d). (ii) Find P(Y=5).

[Marks: 1]

(d). (iii) Find E(Y).

[Marks: 1]

Throw the die until you get each score at least once. Let Z= number of throws.

(e). (i) Find P(Z=6).

[Marks: 3]

(e). (ii) Find P(Z=7).

[Marks: 3]

(e). (iii) Find E(Z).

[Marks: 4]
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