Problem 30 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Fifteen coupons are numbered \(1,2,3, \ldots 15\). Seven coupons are selected at random one at a time withreplacement. The probability that the largest number appearing on the selected coupon is 9 , is (A) \(\left(\frac{9}{16}\right)^{6}\) (B) \(\left(\frac{8}{15}\right)^{7}\) (C) \(\left(\frac{3}{5}\right)^{7}\) (D) none of these

Step-by-Step Solution

Answer

The correct probability expression is none of the given options.

Step 1: Understand the Problem

We are given 15 coupons numbered 1 through 15. We select 7 coupons randomly with replacement, and we want to find the probability that the largest number on these selected coupons is 9. This means that all the drawn numbers are 9 or less, and at least one of them is exactly 9.

Step 2: Calculate Total Outcomes

Since the selection is with replacement, for each draw, we have 15 possible outcomes. We are selecting 7 coupons, so the total number of outcomes is \(15^7\).

Step 3: Calculate Desired Outcomes

The largest number being 9 implies that each of the 7 selected numbers must be no greater than 9. The number of outcomes where all selected numbers are from the set \(\{1, 2, ..., 9\}\) is \(9^7\). Within these, the largest cannot be less than 9, so at least one number must be exactly 9.

Step 4: Use Complementary Counting

Calculate the cases where the largest number is less than 9, meaning all numbers are from 1 to 8. The number of such outcomes is \(8^7\). Subtract these from the total cases of numbers being from 1 to 9.

Step 5: Calculate Probability

The probability that the largest number is exactly 9 is given by \[ \frac{\text{Number of desired outcomes}}{\text{Total outcomes}} = \frac{9^7 - 8^7}{15^7} \]

Step 6: Simplify Expression

Calculate \(9^7 - 8^7\) and divide by \(15^7\) to check if this matches any of the given options.

Key Concepts

Combinatorics

Combinatorics is the study of counting and arranging objects. In probability, it helps us determine the number of ways certain events can occur. Imagine you have a set of items, and you want to select some of them in specific ways. This is where combinatorics becomes crucial.

In our problem, we are selecting 7 coupons from a set of 15, which involves combinations with repetition. Since each draw is independent due to replacement, each selection from the set of 15 is like resetting the process, keeping the chance the same for each draw.

To find out how many ways we can choose these coupons, we use the formula for arrangements with repetition, which is simply raising the number of choices (15 coupons) to the power of the number of selections (7 coupons), i.e., 15 raised to the power of 7. This gives us the total number of possible outcomes for the selection process.

Replacements in probability

Replacements in probability refer to a scenario where each time an item is chosen, it is 'replaced' before the next selection is made. This means after every draw, the pool of items looks exactly the same.

For our coupon problem, this concept plays out in a way that each coupon has the same chance of being selected at every draw. The probability remains constant because every time we draw a coupon, we're drawing from the full set of 15 coupons again.

By replacing a coupon back into the set after it's been drawn, the next draw is unaffected by the previous outcome, maintaining the independence of each event. This is crucial because it allows us to calculate the probability of each outcome straightforwardly, knowing all conditions remain unchanged across draws.

Coupon selection problem

The coupon selection problem involves drawing coupons with the intent to determine a particular outcome regarding their numbers. In this case, you need to ensure that the largest number appearing on the drawn coupons is 9.

To understand the solution, imagine that the solution requires the largest number among the selected coupons to be exactly 9, with all other numbers being less than or equal to 9. Thus, this outcome hinges on a mix of ensuring that numbers are drawn from 1 to 9, but with at least one draw exactly returning a 9.

Here, using complementary counting, we calculate outcomes where the largest number is less than 9 (meaning all numbers are 1 to 8) and subtract these from total outcomes where every number is at most 9. This kind of approach ensures we properly account for all relevant cases to determine the probability correctly. The ultimate step involves simplifying the expression to align with one of the proposed options in the problem, providing an elegant verification of the solution.