Problem 69 - Calculus for Biology and Medicine

Question

An urn contains one black and 14 white balls. Balls are drawn at random, one at a time, until the black ball is selected. Each ball is replaced before the next ball is drawn. Find the probability that at least 20 draws are needed.

Step-by-Step Solution

Answer

The probability that at least 20 draws are needed is \( \left(\frac{14}{15}\right)^{19} \).

Step 1: Understanding the Problem

The problem involves drawing balls from an urn containing 1 black and 14 white balls, with replacement after each draw. Our task is to find the probability of requiring at least 20 draws to draw the black ball.

Step 2: Identify Probability of Drawing the Black Ball

Since there is 1 black ball and 14 white balls, the probability of drawing the black ball in one draw is: \( P(\text{Black}) = \frac{1}{15} \).

Step 3: Identify Probability of Drawing a White Ball

The probability of drawing a white ball in one draw is given by \( P(\text{White}) = \frac{14}{15} \).

Step 4: Use Geometric Distribution

To have at least 20 draws before the black ball is drawn, the first 19 draws must result in a white ball. The geometric distribution is appropriate here, as it models the number of Bernoulli trials before a success.

Step 5: Calculate Probability of 19 Consecutive White Balls

The probability of drawing 19 consecutive white balls is given by \( \left(\frac{14}{15}\right)^{19} \).

Step 6: Compute the Probability of Drawing the Black Ball on the 20th Draw

The probability of drawing the black ball on the 20th draw is calculated using the previous result: \( P(\text{Black on 20th}) = \left(\frac{14}{15}\right)^{19} \times \frac{1}{15} \).

Step 7: Find Probability of At Least 20 Draws

By definition of geometric distribution, the probability that at least 20 draws are needed is the same as having 19 consecutive draws with a white ball, which is \( \left(\frac{14}{15}\right)^{19} \).

Key Concepts

Probability

Probability in this context refers to the likelihood of an event happening when drawing balls from an urn. Here, we're examining the probability of different outcomes for black and white balls. Understanding these likelihoods helps us calculate complex occurrences, like how many draws are needed to finally get a black ball.

- The probability is expressed as a fraction, which represents the number of favorable outcomes over the total possible outcomes.
- In this scenario, since there is only 1 black ball out of 15 total balls, the probability of drawing a black ball in a single draw is:\[P(\text{Black}) = \frac{1}{15}\]- Similarly, the probability of getting a white ball, with 14 white balls in the urn, is:\[P(\text{White}) = \frac{14}{15}\]These basic probabilities become building blocks for understanding more complex problems like ensuring the first 19 draws are not a black ball.

Bernoulli Trials

A Bernoulli trial is a random experiment where there are only two possible outcomes, often termed success or failure. In the problem of drawing balls from an urn, each draw can be considered a Bernoulli trial.

- Here, a 'success' can be defined as drawing a black ball, while a 'failure' is drawing a white ball.
- The probability of success for each trial is \( \frac{1}{15} \), corresponding to pulling a black ball.
- Bernoulli trials are independent, meaning the outcome of one trial does not affect the next. This independence is assured by replacing the ball after each draw.

When we talk about multiple consecutive Bernoulli trials, such as needing several "failures" (white balls) before the first "success" (black ball), we delve into the realm of geometric distribution, which models such scenarios.

Replacement Drawing

Replacement drawing is a technique that ensures the probabilities remain constant throughout the experiment. In each draw, the selected ball is replaced back into the urn, maintaining the original number of each type of ball.

- This means the probability of drawing a black or white ball remains \( \frac{1}{15} \) for black and \( \frac{14}{15} \) for white, throughout the draws.
- Replacement is crucial for maintaining the conditions necessary for a Bernoulli trial. Without replacement, the probabilities would change based on the draws.
- For large numbers of draws, especially when requiring a certain number of failures before success, maintaining these constant probabilities is key for solving the problem effectively.

Without replacement, each draw would depend on all previous draws, greatly complicating calculations and deviating from a Bernoulli framework.

Consecutive Outcomes

In this problem, we're interested in consecutive outcomes, specifically consecutive draws until a specific event occurs. The event we're analyzing is drawing the black ball.

- We require at least 19 consecutive draws resulting in a white ball before potentially drawing a black ball on the 20th draw.
- The probability of this happening corresponds to the probability of all drawn whites before finally getting one black.- This involves a geometric series calculation where each draw is independent, but all must sequence in a specific order until the desired outcome is achieved.The probability of obtaining 19 successive white ball draws followed by a black ball draw employs the geometric distribution:\[P(\text{19 Whites then Black}) = \left(\frac{14}{15}\right)^{19} \times \frac{1}{15}\]Consecutive outcomes are fundamental in probabilistic scenarios like these, both constraining and defining the probability space of possible events.