Question

A coin is weighted deliberately so that the probability of tossing heads is twice the probability of tossing tails. What is the probability of tossing three heads in a row? What is the probability of tossing three tails in a row? What is the relative probability of tossing three heads versus three tails?

Short answer

The probability of tossing three heads is \( \frac{8}{27} \), three tails is \( \frac{1}{27} \), and the relative probability is 8.

Step-by-step solution

Identify Probabilities

Let's denote the probability of tossing heads as \( P(H) \) and the probability of tossing tails as \( P(T) \). According to the problem, \( P(H) = 2 \cdot P(T) \). Since these are the only two outcomes, we also have: \( P(H) + P(T) = 1 \).

Solve for Individual Probabilities

We can set up the equations: \( P(H) = 2P(T) \) and \( P(H) + P(T) = 1 \). Substituting the first equation into the second gives \( 2P(T) + P(T) = 1 \), which simplifies to \( 3P(T) = 1 \). Thus, \( P(T) = \frac{1}{3} \) and \( P(H) = \frac{2}{3} \).

Calculate Probability of Three Heads

The probability of tossing three heads in a row is \( (P(H))^3 \). Substituting the probability from the previous step gives: \[ (\frac{2}{3})^3 = \frac{8}{27} \].

Calculate Probability of Three Tails

The probability of tossing three tails in a row is \( (P(T))^3 \). Substituting the probability from step 2 gives: \[ (\frac{1}{3})^3 = \frac{1}{27} \].

Determine Relative Probability of Three Heads vs. Three Tails

To find the relative probability of tossing three heads versus three tails, divide the probability of tossing three heads by the probability of tossing three tails: \[ \frac{\frac{8}{27}}{\frac{1}{27}} = 8 \].

Key concepts

Weighted Coin

A weighted coin is a special type of coin that does not follow the typical 50-50 chance of landing on heads or tails. Instead, it has a deliberate bias towards one outcome over the other. The bias is usually set for specific purposes, like in the exercise problem where the probability of landing on heads is twice that of tails. This kind of setup is often used to illustrate probability concepts that are not as straightforward as a fair coin toss.

• The sum of the probabilities of landing on either side (heads or tails) always equals 1.
• If you know one probability in a weighted coin, you can easily find the other by subtraction.
• Weighted coins can make probability questions more challenging, offering deeper insights into how probability can be manipulated.

Weighted coins provide an excellent means for understanding probability theory beyond the basics. They show that outcomes can be stretched to favor one side, changing the entire probability landscape. This manipulative feature of weighted coins is crucial in various real-life scenarios, from gaming to statistical modeling.

Heads and Tails

When discussing heads and tails in probability, most immediately think of a fair coin with an equal chance of landing either way. However, things change when considering a weighted or biased coin. In our example, the coin is weighted such that the chance of landing heads, denoted as \( P(H) \), is twice the probability of tails, \( P(T) \).

• In our example, if \( P(T) = \frac{1}{3}\), then \( P(H) \) must be \( 2 \times \frac{1}{3} = \frac{2}{3} \).
• The probabilities must always add up to 1: \( P(H) + P(T) = 1 \).
• Weighted coins show how shifting the likelihood of heads or tails alters expected outcomes in repeated trials.

Understanding how these adjustments affect the outcomes is vital in mastering probability concepts. Recognizing that not all coins are 'equal' opens the door to exploring more complex probability issues, such as the one presented in this exercise.

Relative Probability

Relative probability compares the likelihood of different outcomes. In our exercise, we wanted to compare the chance of getting three heads in a row against getting three tails in a row. This comparison is crucial because it tells us how one outcome is weighted relative to another.

• The probability of three heads in a row, given our coin, is \( \left(\frac{2}{3}\right)^3 = \frac{8}{27} \).
• For three tails, it's \( \left(\frac{1}{3}\right)^3 = \frac{1}{27} \).
• The relative probability is determined by dividing the probability of heads by the probability of tails: \( \frac{8/27}{1/27} = 8 \).

Why is this useful? Such comparisons allow us to see how much more likely one series of events is compared to another. Relative probability simplifies understanding by reducing probabilities into ratios easy to understand and interpret. It’s especially instrumental in decision-making processes where choosing the more probable outcome is key.