Solved: Arianne plays a game of darts. The distance that her darts land from the centre,... [IB Mathematics Applications & Interpretation (Math AI)]

Question

HLPaper 2

Arianne plays a game of darts. The distance that her darts land from the centre, O, of the board can be modelled by a normal distribution with mean 10 cm and standard deviation 3 cm.

Find the probability that a dart lands less than 13 cm from O.

[2]

Verified

Solution

Let $X$ be the random variable "distance from O ".

$X \sim \mathrm{N}(10,3^2)$

$\mathrm{P}(X<13)=0.841(0.841344 \ldots)$

(M1)A1

Find the probability that a dart lands more than 15 cm from O.

[2]

Verified

Solution

$(\mathrm{P}(X>15)=) 0.0478(0.0477903) \quad \boldsymbol{A1}$

Each of Arianne's throws is independent of her previous throws. Find the probability that Arianne throws two consecutive darts that land more than 15 cm from O.

[2]

Verified

Solution

$\mathrm{P}(X>15) \times \mathrm{P}(X>15)$ (M1) $=0.00228(0.00228391 \ldots)$ A1

In a competition a player has three darts to throw on each turn. A point is scored if a player throws all three darts to land within a central area around O. When Arianne throws a dart the probability that it lands within this area is 0.8143. Find the probability that Arianne does not score a point on a turn of three darts.

[2]

Verified

Solution

$1-(0.8143)^{3}$ M1A1

In the competition Arianne has ten turns, each with three darts. Find Arianne's expected score in the competition.

[2]

Verified

Solution

let $Y$ be the random variable "number of points scored" evidence of use of binomial distribution (M1) $(\mathrm{E}(Y)=) 10 \times 0.539949 \ldots$ $=5.40$ A1

Find the probability that Arianne scores at least 5 points in the competition.

[3]

Verified

Solution

$(\mathrm{P}(Y \geq 5)=) 0.717(0.716650 \ldots)$

Find the probability that Arianne scores at least 5 points and less than 8 points.

[2]

Verified

Solution

$\mathrm{P}(5 \leq Y<8)$ (M1) $=0.628(0.627788 \ldots)$ A1 Note: Award $\boldsymbol{M1}$ for a correct probability statement or indication of correct lower and upper bounds, 5 and 7.

Given that Arianne scores at least 5 points, find the probability that Arianne scores less than 8 points.

[2]

Verified

Solution

$\frac{\mathrm{P}(5 \leq Y<8)}{\mathrm{P}(Y \geq 5)}\left(=\frac{0.627788 \ldots}{0.716650 \ldots}\right)$ (M1) $=0.876(0.876003 \ldots)$ A1

Arianne plays a game of darts. The distance that her darts land from the centre, O, of the board can be modelled by a normal distribution with mean 10 cm and standard deviation 3 cm.

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