We are throwing darts on a disk-shaped board of radius 5. We assume that the proposition of the dart is a uniformly chosen point in the disk

Question

We are throwing darts on a disk-shaped board of radius 5. We assume that the proposition of the dart is a uniformly chosen point in the disk. The board has a disk-shaped bullseye with radius 1. Suppose that we throw a dart 2000 times at the board. Estimate the probability that we hit the bullseye at least 100 times.

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Sapo 3 years 2021-08-03T10:15:01+00:00 1 Answers 154 views 0

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  1. Thunguyet
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    2021-08-03T10:16:14+00:00
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    Answer:

    the probability that we hit the bullseye at least 100 times is 0.0113

    Step-by-step explanation:

    Given the data in the question;

    Binomial distribution

    We find the probability of hitting the dart on the disk

    ⇒ Area of small disk / Area of bigger disk

    ⇒ πR₁² / πR₂²

    given that; disk-shaped board of radius R² = 5, disk-shaped bullseye with radius R₁ = 1

    so we substitute

    ⇒ π(1)² / π(5)² = π/π25 = 1/25 = 0.04

    Since we have to hit the disk 2000 times, we represent the number of times the smaller disk ( BULLSEYE ) will be hit by X.

    so

    X ~ Bin( 2000, 0.04 )

    n = 2000

    p = 0.04

    np = 2000 × 0.04 = 80

    Using central limit theorem;

    X ~ N( np, np( 1 – p ) )

    we substitute

    X ~ N( 80, 80( 1 – 0.04 ) )

    X ~ N( 80, 80( 0.96 ) )

    X ~ N( 80, 76.8 )

    So, the probability that we hit the bullseye at least 100 times, P( X ≥ 100 ) will be;

    we covert to standard normal variable

    ⇒ P( X ≥  \frac{100-80}{\sqrt{76.8} } )

    ⇒ P( X ≥ 2.28217 )

    From standard normal distribution table

    P( X ≥ 2.28217 ) = 0.0113

    Therefore, the probability that we hit the bullseye at least 100 times is 0.0113

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