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Chapter 11: Problem 26

A fair coin is tossed two times in succession. The set of equally likely outcomes is \(\\{H H, H T, T H, T T\\}\). Find the probability of getting at least one head.

Short Answer

Expert verified
The probability of getting at least one head when flipping a fair coin twice in succession is \(\frac{3}{4}\).

Step by step solution

01

List all possible outcomes

A fair coin has two outcomes: Heads (H) or Tails (T). If the coin is flipped twice, there are 4 equally likely outcomes: \(\{HH, HT, TH, TT\}\).
02

Determine favourable outcomes

To find the probability of getting at least one head, identify the outcomes that include one or more heads. The favorable outcomes are \(\{HH, HT, TH\}\).
03

Calculate the Probability

The probability of an event is given by the ratio of the number of favourable outcomes to the total number of outcomes. There are 3 favourable outcomes and 4 possible outcomes. Therefore, the probability is \(\frac{3}{4}\).

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Key Concepts

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

Understanding a Fair Coin
A fair coin is a coin that has an equal likelihood of landing on either side when tossed.
This means there is a 50% chance of landing on heads (H) and a 50% chance of landing on tails (T).
Fairness is essential as it ensures that no side of the coin is favored when flipped.
  • 50% chance for heads (H)
  • 50% chance for tails (T)
In probability exercises, assuming that a coin is fair allows us to calculate outcomes with precision. It's crucial to recognize the fairness of tools in probability to ensure accurate results.
Possible Outcomes of Coin Tosses
When a fair coin is tossed two times in a row, each toss is independent of the other.
This independence results in multiple outcomes as a sequence of tosses becomes possible.
  • First toss outcomes: H or T
  • Second toss outcomes: H or T
By combining these tosses, you have:
  • Heads on both tosses (HH)
  • Heads first, then tails (HT)
  • Tails first, then heads (TH)
  • Tails on both tosses (TT)
Notice how with two tosses, there are four distinct outcomes, each equally likely due to the coin's fair nature.
At Least One Head
The phrase 'at least one head' refers to the scenarios where one or more of the coin tosses result in a head.
In our set of possible outcomes
  • HH
  • HT
  • TH
  • TT
All combinations except TT have at least one head. Therefore, the favourable outcomes we are interested in for this scenario are
  • HH
  • HT
  • TH
It is crucial in probability to identify all configurations that meet the criteria of 'at least one'.
Calculating the Probability
Probability is the measure of how likely an event is to occur.
It is calculated by dividing the number of favourable outcomes by the total number of possible outcomes.
This gives us a sense of how often we can expect a certain outcome when the process is repeated many times.For our coin toss situation, the probability of getting at least one head is derived as follows:
  • Total possible outcomes: 4 (HH, HT, TH, TT)
  • Favorable outcomes with at least one head: 3 (HH, HT, TH)
Therefore, the probability is \[\frac{3}{4} = 0.75\]This calculation indicates a 75% likelihood that at least one head will appear when a fair coin is tossed twice.

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Most popular questions from this chapter

Involve computing expected values in games of chance. The spinner on a wheel of fortune can land with an equal chance on any one of ten regions. Three regions are red, four are blue, two are yellow, and one is green. A player wins \(\$ 4\) if the spinner stops on red and \(\$ 2\) if it stops on green. The player loses \(\$ 2\) if it stops on blue and \(\$ 3\) if it stops on yellow. What is the expected value? What does this mean if the game is played ten times?

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