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Chapter 3: Problem 5

Coin flips. If you flip a fair coin 10 times, what is the probability of (a) getting all tails? (b) getting all heads? (c) getting at least one tails?

Short Answer

Expert verified
(a) 0.0009765625; (b) 0.0009765625; (c) 0.9990234375.

Step by step solution

01

Understand the problem

The problem involves determining the probability of certain outcomes when flipping a fair coin 10 times. Here, 'fair' means that each flip has an equal chance of being heads or tails, specifically, a probability of 0.5 for heads and 0.5 for tails.
02

Calculate the probability of all tails

To find the probability of getting all tails in 10 flips, we need to consider that each flip is an independent event with a probability of 0.5. Thus, the probability of getting tails in each flip is 0.5. The probability of getting tails for 10 consecutive flips is calculated by multiplying the probability for each flip: \[ P( ext{all tails}) = (0.5)^{10} = \text{0.0009765625} \].
03

Calculate the probability of all heads

Similarly, the probability of getting all heads is the same as getting all tails, since the coin is fair. So, the calculation is \[ P( ext{all heads}) = (0.5)^{10} = \text{0.0009765625} \].
04

Calculate the probability of at least one tails

To find the probability of getting at least one tail in 10 flips, it's easier to calculate the probability of the complementary event鈥攇etting no tails (which is the same as getting all heads)鈥攁nd subtracting from 1. Using the complement rule: \[ P( ext{at least one tails}) = 1 - P( ext{all heads}) \]. Since we already calculated \( P( ext{all heads}) = 0.0009765625 \) from Step 3, \[ P( ext{at least one tails}) = 1 - 0.0009765625 = \text{0.9990234375} \].

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Key Concepts

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

Independent Events
When dealing with probability, it's essential to grasp the concept of independent events. Events are considered independent if the outcome of one event does not affect the outcome of another. In the context of coin flips, each flip represents an independent event. Whether the coin lands on heads or tails on one flip has no impact on how it will land on subsequent flips.

A simple way to think about independent events is to consider the nature of a fair coin. Every flip is isolated, meaning:
  • Each flip has a fixed probability of 0.5 for heads and 0.5 for tails.
  • The outcome of one flip doesn't change the probability of the next flip.
Therefore, when calculating probabilities involving multiple flips, we multiply the probabilities of individual flips, because they are independent events.
Complement Rule
The complement rule is a handy tool in probability theory, especially for complex events. It helps us find the probability of an event by calculating the probability of the event's complement and then subtracting that from 1.

For example, if you want to find the probability of getting at least one tails in 10 coin flips, it can be easier to first figure out the probability of the opposite event: getting no tails (or all heads). The complement rule informs us that:
  • The probability of at least one tails = 1 minus the probability of all heads.
This approach often simplifies calculations and reduces potential errors, particularly in scenarios involving multiple trials or outcomes.
Basic Probability Calculations
Basic probability calculations are fundamental to understanding how likely events are to occur. Using simple principles, you can determine the probability of various events, ensuring that your logic remains sound and consistent.

For single-event probability, such as a coin flip, you choose between two outcomes. Since it's a fair coin, each outcomes chance is 0.5. If you want to calculate the probability of an event involving multiple outcomes, like getting 10 tails in a row, you multiply the probability of each individual event:
  • For 10 tails in a row: \[ P(\text{10 tails}) = (0.5)^{10} = 0.0009765625 \].
This multiplication rule stems from the concept of independent events and ensures consistency across calculations in probability exercises.

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Most popular questions from this chapter

A portfolio's value increases by \(18 \%\) during a financial boom and by \(9 \%\) during normal times. It decreases by \(12 \%\) during a recession. What is the expected return on this portfolio if each scenario is equally likely?

The game of European roulette involves spinning a wheel with 37 slots: 18 red, 18 black, and 1 green. A ball is spun onto the wheel and will eventually land in a slot, where each slot has an equal chance of capturing the ball. Gamblers can place bets on red or black. If the ball lands on their color, they double their money. If it lands on another color, they lose their money. (a) Suppose you play roulette and bet \(\$ 3\) on a single round. What is the expected value and standard deviation of your total winnings? (b) Suppose you bet \(\$ 1\) in three different rounds. What is the expected value and standard deviation of your total winnings? (c) How do your answers to parts (a) and (b) compare? What does this say about the riskiness of the two games?

Below are four versions of the same game. Your archnemesis gets to pick the version of the game, and then you get to choose how many times to flip a coin: 10 times or 100 times. Identify how many coin flips you should choose for each version of the game. It costs \(\$ 1\) to play each game. Explain your reasoning. (a) If the proportion of heads is larger than \(0.60,\) you win \(\$ 1\). (b) If the proportion of heads is larger than 0.40 , you win \(\$ 1\). (c) If the proportion of heads is between 0.40 and 0.60 , you win \(\$ 1\). (d) If the proportion of heads is smaller than \(0.30,\) you win \(\$ 1\).

Determine if the statements below are true or false, and explain your reasoning. (a) If a fair coin is tossed many times and the last eight tosses are all heads, then the chance that the next toss will be heads is somewhat less than \(50 \%\). (b) Drawing a face card (jack, queen, or king) and drawing a red card from a full deck of playing cards are mutually exclusive events. (c) Drawing a face card and drawing an ace from a full deck of playing cards are mutually exclusive events.

\(\mathrm{P}(\mathrm{A})=0.3, \mathrm{P}(\mathrm{B})=0.7\) (a) Can you compute \(\mathrm{P}(\mathrm{A}\) and \(\mathrm{B})\) if you only know \(\mathrm{P}(\mathrm{A})\) and \(\mathrm{P}(\mathrm{B}) ?\) (b) Assuming that events \(A\) and \(B\) arise from independent random processes, i. what is \(\mathrm{P}(\mathrm{A}\) and \(\mathrm{B})\) ? ii. what is \(\mathrm{P}(\mathrm{A}\) or \(\mathrm{B}) ?\) iii. what is \(\mathrm{P}(\mathrm{A} \mid \mathrm{B}) ?\) (c) If we are given that \(\mathrm{P}(\mathrm{A}\) and \(\mathrm{B})=0.1,\) are the random variables giving rise to events \(\mathrm{A}\) and \(\mathrm{B}\) independent? (d) If we are given that \(\mathrm{P}(\mathrm{A}\) and \(\mathrm{B})=0.1,\) what is \(\mathrm{P}(\mathrm{A} \mid \mathrm{B}) ?\)
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