91Ó°ÊÓ

For eight coins being tossed, the probabilities of getting various numbers of heads are shown in the table. Use the addition principle to find the probability of each event indicated below. $$\begin{array}{|l|l|l|l|l|l|} \hline \text { Number of heads } & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Probability } & \frac{1}{286} & \frac{8}{256} & \frac{28}{286} & \frac{56}{256} & \frac{70}{226} \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|} \hline \text { Number of heads } & 5 & 6 & 7 & 8 \\ \hline \text { Probability } & \frac{4}{28} & \frac{24}{256} & \frac{1}{28} & \frac{1}{26} \\ \hline \end{array}$$ (a) three or fewer heads (b) more heads than tails (c) all heads or all tails (d) an odd number of heads

Step by step solution

01

Calculate the Probability of Three or Fewer Heads

To find the probability of three or fewer heads, add up the probability of having 0, 1, 2, or 3 heads: \(P = \frac{1}{286} + \frac{8}{256} + \frac{28}{286} + \frac{56}{256}\)

02

Calculate the Probability of More Heads than Tails

More heads than tails means getting 5, 6, 7, or 8 heads. Again, sum up the corresponding probabilities: \(P = \frac{4}{28} + \frac{24}{256} + \frac{1}{28} + \frac{1}{26}\)

03

Calculate the Probability of All Heads or All Tails

For all heads or all tails, look up the probability of 0 and 8 heads. \(P = \frac{1}{286} + \frac{1}{26}\)

04

Calculate the Probability of an Odd Number of Heads

An odd number of heads means getting 1, 3, 5, or 7 heads. Sum up the matching probabilities: \(P = \frac{8}{256} + \frac{56}{256} + \frac{4}{28} + \frac{1}{28}\)

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Key Concepts

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

Addition Principle

The addition principle is all about combining the probabilities of mutually exclusive events. In probability, two events are mutually exclusive when they cannot occur at the same time.
For example, rolling a die can result in either a 1, 2, 3, 4, 5, or 6, but not two of these outcomes simultaneously.

• When asked to find the probability of multiple outcomes, simply add the probabilities of each individual outcome.
• This principle is crucial when dealing with situations where more than one outcome satisfies a condition.

In the context of coin tossing, if you want to find the probability of getting a certain number of heads, you might need to sum the probabilities for several different outcomes. Using the addition principle, you can seamlessly account for all favorable events.

Probability Calculations

Probability calculations involve determining the likelihood that a particular event will occur. These calculations can range from simple situations to more complex scenarios.
Calculating probabilities often involves:

• Identifying all possible outcomes.
• Pinpointing the favorable outcomes associated with the event.
• Using the ratio of favorable outcomes to total possible outcomes to find the probability.

In cases where there are multiple favorable outcomes, as shown in the original exercise, probabilities can be added together using the addition principle. For example, to find the probability of getting three or fewer heads when tossing eight coins, you add the probabilities of getting 0, 1, 2, and 3 heads.

Coin Tossing Experiments

Coin tossing experiments are a classic example of probability. Each toss is an independent event with two possible outcomes - heads or tails.
This makes the probability calculation straightforward:

• The probability of getting heads is 0.5.
• The probability of getting tails is also 0.5.

However, with multiple coin tosses, the number of potential outcomes grows exponentially. For instance, tossing eight coins results in 256 possible combinations of heads and tails.

• Each combination can occur in different sequences, which affects the overall probability calculations.
• Understanding how to build sample spaces (all potential outcomes) is crucial to solving these problems.

In this framework, outcomes like getting an odd number of heads or all tails can be explored using the addition principle to sum the probabilities of favorable sequences.