Question 8 of 55intermediate🔧 ApplyNumerical3 marks

In a lottery, a person chooses six different natural numbers at random from 1 to 20, and if these six numbers match with the six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game? [Hint order of the numbers is not important.]

Correct Answer

The probability of winning the prize is 138760\frac{1}{38760}.

Exercise: EXERCISE 14.2 | Q: 11 | (Chapter: Page 17)
For More Understanding

Explanation

This question applies the concept of combinations to probability. Since the hint explicitly states that the order of numbers is not important, we determine the total number of possible outcomes using the combination formula nCr^nC_r. The total number of ways to choose 6 numbers from 20 is 20C6^{20}C_6. There is only 1 favorable outcome where the chosen numbers match the committee's fixed numbers. The probability is then calculated as the ratio of favorable outcomes to total outcomes.

Solution Steps

  1. Step 1: Determine Total Number of Outcomes The person chooses 6 different numbers from 1 to 20. Since the order is not important, the total number of ways to choose the numbers is calculated using combinations: Total outcomes=20C6=20!6!(20−6)!=20×19×18×17×16×156×5×4×3×2×1=38760\text{Total outcomes} = ^{20}C_6 = \frac{20!}{6!(20-6)!} = \frac{20 \times 19 \times 18 \times 17 \times 16 \times 15}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 38760

  3. Step 2: Determine Number of Favorable Outcomes For the person to win, their 6 numbers must match the 6 numbers fixed by the lottery committee. There is only 1 way for this to happen. Favorable outcomes=1\text{Favorable outcomes} = 1

  5. Step 3: Calculate Probability Using the probability formula P(E)=Number of favorable outcomesTotal number of possible outcomesP(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}: P(Winning)=138760P(\text{Winning}) = \frac{1}{38760}

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