CAIE A-Level Mathematics AS 5.4.2 Binomial and Geometric Distributions

[Maximum number: 6]

Salah decides to attempt the crossword puzzle in his newspaper each day. The probability that he will complete the puzzle on any given day is 0.65 , independent of other days.

(a)

Find the probability that Salah completes the puzzle for the first time on the 5th day.

[ 1 ]

To complete the puzzle for the first time on day 5, Salah must fail on the first four days and succeed on day 5:
$P=0.35^4(0.65)=0.00975.$

(b)

Find the probability that Salah completes the puzzle for the second time on the 5th day.

[ 2 ]

For the second success to occur on day 5, there must be exactly one success in the first four days, followed by a success on day 5:
$P=\binom41(0.35)^3(0.65)^2=0.0725.$

(c)

Find the probability that Salah completes the puzzle fewer than 5 times in a week ( 7 days).

[ 3 ]

Let X be the number of completed puzzles in 7 days. Then $X\sim\operatorname{Bin}(7,0.65)$ . Hence,
$\begin{aligned} P(X<5) &=1-P(X\geq5)\\ &=1-\left[\binom75(0.65)^5(0.35)^2 +\binom76(0.65)^6(0.35)+0.65^7\right]\\ &=0.468. \end{aligned}$

CAIE A-Level Mathematics AS 5.4.2 Geometric Distribution Questions

Exam points

CAIE A-Level Mathematics AS 5.4.2 Geometric Distribution Questions question 1

Question (a)

Question (b)

Question (c)