A wheel is divided into 10 equal sectors numbered 1 through 10. In an initial experiment, … — JAMB UTME

A wheel is divided into $10$ equal sectors numbered $1$ through $10$ . In an initial experiment, the wheel is spun $400$ times, and a prime number is recorded $180$ times. Additional spins are to be conducted, all of which result in non-prime numbers. How many additional spins must be performed so that the overall experimental relative frequency of landing on a prime number equals the theoretical probability of landing on a prime number?

$50$ Cevap
B
$100$
C
$200$
D
$450$

Cevap

50 additional spins

The correct value is 50. The theoretical probability of landing on a prime number from 1 to 10 is 4/10 = 2/5, since the prime numbers are 2, 3, 5, and 7. With 180 prime outcomes in 400 spins, adding N non-prime spins yields an experimental relative frequency of 180 / (400 + N). Setting 180 / (400 + N) = 2/5 yields 800 + 2N = 900, giving N = 50.

Adım Adım Çözüm

Determine the theoretical probability of landing on a prime number.

The prime numbers between

1

and

10

are

2, 3, 5,

and

7

. Thus, there are

4

favorable outcomes out of

10

, giving

P(\text{theoretical}) = \frac{4}{10} = \frac{2}{5}

Theoretical probability is the ratio of favorable outcomes to the total number of equally likely outcomes.

Formulate the expression for experimental relative frequency after

N

additional non-prime spins.

The number of prime occurrences remains

180

, while the total number of spins becomes

400 + N

. Thus,

P(\text{experimental}) = \frac{180}{400 + N}

Since all

N

additional spins produce non-prime numbers, the count of prime outcomes does not increase, but the total number of trials increases by

N

Equate the experimental relative frequency to the theoretical probability and solve for

N

\frac{180}{400 + N} = \frac{2}{5} \implies 2(400 + N) = 180 \times 5 \implies 800 + 2N = 900 \implies 2N = 100 \implies N = 50.

Setting the experimental relative frequency equal to the theoretical probability allows determining the exact number of additional non-prime spins needed.

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Theoretical vs Experimental Probability

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