Question

A box contains cards numbered 11 to 123. A card is drawn at random from the box. Find the probability that the number on the drawn card
A. a square number
B. a multiple of 7

Answer 1

Hint: First of all, find the square numbers between 11 to 123 and find the total number of possible outcomes of getting a square number. Then find the multiples between 11 to 123 and find the total number of possible outcomes of getting a multiple of 7 to find its probability.

Complete step-by-step answer:
Given numbers on card \[ = 11,12,13,14,................................,123\]
Total number of cards \[ = 123 - 11 + 1 = 113\]
A. a square number
We know that, square of numbers \[ = {1^2},{2^2},{3^2},{4^2},{5^2},{6^2},{7^2},{8^2},{9^2},{10^2},{11^2},{12^2},.............\]
                                                              \[ = 1,4,9,16,25,36,79,64,81,100,121,144,......................\]
Square numbers between 11 to 123 are \[16,25,36,79,64,81,100,121\]
So, number of square numbers between 11 to 123 = 8
We know that the probability of an event is given by \[P\left( {\text{E}} \right) = \dfrac{{{\text{Total number of possible outcomes}}}}{{{\text{Total number of outcomes}}}}\]
Here, total number of outcomes = total number of cards = 113
Total number of possible outcomes = number of square numbers between 11 to 123 = 8
Thus, the probability of getting a square number \[ = \dfrac{8}{{113}}\]
B. a multiple of 7
We know that the multiples of 7 are \[7,14,21,28,35,42,49,56,63,70,77,84,91,98,105,112,119,126,...............\]
So, number of multiples of 7 between 11 to 123 = 16
Here, total number of outcomes = total number of cards = 113
Total number of possible outcomes = number of multiples of 7 between 11 to 123 = 16
Thus, the probability of getting a multiple of 7 \[ = \dfrac{{16}}{{113}}\]

Note: The number of possible outcomes is always greater than or equal to the total number of outcomes. The probability of an event is always less than or equal to one and greater than or equal to zero.