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Chapter 12: Problem 18

A fair coin is flipped twice. What is the probability of showing heads on both flips?

Short Answer

Expert verified
The probability is \( \frac{1}{4} \).

Step by step solution

01

Identify Possible Outcomes

When a fair coin is flipped twice, the possible outcomes are: HH (heads on both flips), HT (heads on the first flip, tails on the second), TH (tails on the first flip, heads on the second), and TT (tails on both flips). This gives us a total of 4 possible outcomes.
02

Identify Favorable Outcome

The favorable outcome in this problem is obtaining heads on both flips, which corresponds to the outcome HH.
03

Calculate Probability

Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. There is 1 favorable outcome (HH), and there are 4 possible outcomes in total. Therefore, the probability is: \( \frac{1}{4} \).

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Key Concepts

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

Coin Flip
Flipping a coin is a classic example in probability, used to illustrate random outcomes. A standard coin has two sides: heads and tails. Each time you flip it, one of these sides will land face up.
  • The action of flipping can be considered random and fair, assuming the coin is unbiased and the flip is not influenced by external factors.
  • In any given flip, there are only two possible outcomes: heads (H) and tails (T).
Although it seems simple, this basic concept forms the foundation for more complex probability exercises, such as flipping a coin multiple times.
Possible Outcomes
When discussing random events, such as a coin flip, "possible outcomes" refer to all the different results that can occur.
  • When flipping a fair coin twice, we multiply the outcomes of each flip. Each flip has two possible outcomes: heads or tails.
  • With two flips, this creates a total of four possible combinations or outcomes: HH, HT, TH, and TT.
Each of these combinations is equally likely if the coin is fair, meaning they all have an equal chance of occurring. Mapping out these outcomes helps in understanding how to identify and count favorable occurrences.
Favorable Outcomes
In probability, a favorable outcome is any result which meets the criteria set by the problem or question.
  • In the context of our exercise, the favorable outcome is flipping heads both times, represented by the outcome HH.
  • This particular combination is what the problem is looking to calculate the probability for among all possible outcomes.
Identifying and counting the number of favorable outcomes is key, as this number is used as the numerator in calculating probability.
Probability Calculation
The probability of an event is a measure of how likely that event is to occur. In this exercise, the process for calculating probability involves a few straightforward steps:
  • First, count the number of favorable outcomes. In this exercise, that is just one: HH.
  • Next, count the total number of possible outcomes, which we determined to be four: HH, HT, TH, and TT.
  • Finally, divide the number of favorable outcomes by the total number of possible outcomes: \( \frac{1}{4} \).
This fraction, or ratio, represents the probability of getting heads on both coin flips. The result tells us there is a 25% chance, or one in four likelihood, that both flips will result in heads.

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Most popular questions from this chapter

A computer password must be eight characters long. How many passwords are possible if only the 26 letters of the alphabet are allowed?

If you pick one card at random from a standard deck of cards, what is the probability it is not the Ace of Spades?

The table below shows the number of credit cards owned by a group of individuals. If one person was chosen at random, find the probability that the person was male and had two or more credit cards. $$ \begin{array}{|l|l|l|l|l|} \hline & \text { Zero } & \text { One } & \text { Two or more } & \text { Total } \\ \hline \text { Male } & 9 & 5 & 19 & 33 \\ \hline \text { Female } & 18 & 10 & 20 & 48 \\ \hline \text { Total } & 27 & 15 & 39 & 81 \\ \hline \end{array} $$

A math class consists of 25 students, 14 female and 11 male. Two students are selected at random to participate in a probability experiment. Compute the probability that a. a male is selected, then a female. b. a female is selected, then a male. c. two males are selected. d. two females are selected. e. no males are selected.

A friend devises a game that is played by rolling a single six-sided die once. If you roll a 6 , he pays you $$\$ 3$$; if you roll a 5 , he pays you nothing; if you roll a number less than 5 , you pay him $$\$ 1.$$ Compute the expected value for this game. Should you play this game?
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