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A box contains 10 red marbles and 10 green marbles. a. Sampling at random from this box five times with replacement, you have drawn a red marble all five times. What is the probability of drawing a red marble the sixth time? b. Sampling at random from this box five times without replacement, you have drawn a red marble all five times. Without replacing any of the marbles, what is the probability of drawing a red marble the sixth time? c. You have tossed a fair coin five times and have obtained heads all five times. A friend argues that according to the law of averages, a tail is due to occur and, hence, the probability of obtaining a head on the sixth toss is less than \(.50 .\) Is he right? Is coin tossing mathematically equivalent to the procedure mentioned in part a or the procedure mentioned in part b above? Explain.

Step by step solution

01

Part A: With Replacement

When sampling with replacement, the probability remains the same every time a marble is drawn. To find this probability, we divide the number of red marbles (10) by the total number of marbles (20). So P(Red on sixth draw) = \(\frac{10}{20} = 0.5\). Given that we are replacing marbles after each draw, earlier draws do not change this probability.

02

Part B: Without Replacement

When sampling without replacement, the number of available marbles changes after each draw. We started with 10 red marbles out of 20 total marbles. After drawing 5 red marbles, there are now 5 red marbles and 15 total marbles. So P(Red on sixth draw) = \(\frac{5}{15} = 0.33\). Without replacement, the probability changes after each draw.

03

Part C: Coin tossing vs. Drawing marbles

Tossing a coin is an example of independent events, similar to drawing marbles with replacement. Regardless of previous tosses, the probability of getting heads or tails is always 0.5. This means the friend is incorrect, the probability of getting a head on the sixth toss remains 0.5. Therefore, coin tossing is mathematically equivalent to drawing marbles with replacement mentioned in part a, not part b.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sampling with Replacement

When we talk about sampling with replacement, we're referring to a process where each item drawn from a pool is returned before the next draw. In scenarios like drawing marbles from the box in the exercise, this means after you pick a marble, you put it back. So, the total number of marbles remains the same for every draw.

This concept ensures that the probability of selecting a specific type of item remains consistent across all draws. In simpler terms, the chance of drawing a red marble remains at 50% because each time you draw, there are always 10 red and 10 green marbles in the box.

Using replacement makes each event independent of the others. The past draws don't impact future results because after each draw, the conditions return to exactly as they were at the outset.

Sampling without Replacement

Sampling without replacement, on the other hand, changes the game. Here, when you draw an item, you don't put it back. This means your selection impacts the subsequent options available.

Imagine drawing marbles without replacing them; each draw reduces the total number of marbles and possibly the number of red or green marbles in the box. Thus, drawing all red marbles five times affects what remains. In the exercise, after removing 5 red marbles, only 5 red are left with 15 marbles total, changing the probability of the next draw.

This scenario mimics real-life situations where selections are permanent, like dealing cards in a game. The likelihood changes based on what is left, making previous draws influential.

Independent Events

An independent event means the outcome of one event does not affect the outcome of another. Think of flipping a fair coin; each flip is independent.

In the exercise, coin tossing is highlighted as being independent. Regardless of how many times you toss heads, the next event always holds the same probability - 50% heads or 50% tails. Hence, your friend's reasoning on the law of averages is flawed here.

Independent scenarios are similar to sampling with replacement because past events have no bearing on future results. Every toss or draw of marble resets the probability conditions as if starting anew.

Law of Averages

The law of averages is a common misconception that things have to "even out" in the short run. It is often mistakenly used, as in the exercise, where a friend thought a tail was "due" after multiple heads in a row.

In reality, the law of large numbers tells us this might occur over a vast number of trials, where averages start to stabilize. However, in a small number of trials, such as the six flips or draws, independent events mean each outcome remains possible and likely unchanged.

Therefore, coin tossing as described does not predict outcomes based on previous results, nor does any number of draws with replacement. Always remember, the short-term average will not necessarily reflect long-term probability patterns.