In a game, a fair wheel lands on a single integer from 1 to 100, inclusive, with each inte… — GMAT

In a game, a fair wheel lands on a single integer from $1$ to $100$ , inclusive, with each integer being equally likely to be selected. What is the probability that the integer selected is either a prime number or leaves a remainder of $1$ when divided by $5$ , but not both?

$\frac{7}{20}$ Cevap
B
$\frac{17}{50}$
C
$\frac{2}{5}$
D
$\frac{9}{20}$
E
$\frac{1}{20}$

Cevap

The correct answer is 7/20.

The total number of possible outcomes is 100. There are 25 prime numbers in the range 1 to 100, and 20 numbers that leave a remainder of 1 when divided by 5. Exactly 5 numbers (11, 31, 41, 61, 71) meet both conditions. To find the numbers that meet either condition but NOT both, subtract the 5 overlapping numbers from each category: 20 numbers are prime only, and 15 numbers leave a remainder of 1 only. The number of favorable outcomes is 20 + 15 = 35. Therefore, the single-event probability is 35/100 = 7/20.

Adım Adım Çözüm

Determine the total sample space size.

The total number of possible outcomes is

N = 100

Each integer from 1 to 100 inclusive is equally likely.

Identify and count all prime numbers in the set

\{1, 2, \dots, 100\}

There are 25 prime numbers:

\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97\}

Note that 1 is not a prime number.

Identify and count all integers in

\{1, 2, \dots, 100\}

that leave a remainder of 1 when divided by 5.

There are 20 such integers:

\{1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, 81, 86, 91, 96\}

These integers take the form

5k + 1

for integers

k \ge 0

where

1 \le 5k + 1 \le 100

Find the intersection of both sets (numbers that are prime AND leave a remainder of 1 when divided by 5).

The 5 numbers satisfying both conditions are

\{11, 31, 41, 61, 71\}

Any integer leaving a remainder of 1 when divided by 5 must end in 1 or 6. No prime ends in 6. The primes ending in 1 in this range are 11, 31, 41, 61, and 71.

Calculate the number of favorable outcomes satisfying 'either condition, but not both' (symmetric difference).

Favorable outcomes =

(25 - 5) + (20 - 5) = 20 + 15 = 35

We must count numbers that are only prime (20) plus numbers that are only remainder 1 (15), excluding the 5 overlapping numbers.

Calculate the single-event probability.

P = \frac{35}{100} = \frac{7}{20}

Divide favorable outcomes by total sample space size.

Anahtar Kavram

Basic single-event probability with set operations and number properties (prime numbers, remainders).

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