3) Consider the experiment tossing a coin 10 times. a) How many ways can the experiment come up with head occurring exactly 7 times. b) How many possible ways can heads and tails come up? (What is the size of the sample space)
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![### Experiment: Tossing a Coin 10 Times
#### Question 3:
Consider the experiment of tossing a coin 10 times.
a) How many ways can the experiment result in exactly 7 heads?
b) How many possible outcomes are there for the heads and tails? (What is the size of the sample space?)
### Detailed Explanation
**Graph or Diagram Explanation:**
There is no graph or diagram in this content. However, the explanation involves combinatorial mathematics, often represented using Pascal's Triangle or binomial coefficients.
**Solution Approach:**
1. **Part (a): Combinatorial Calculation**
- To determine the number of ways the experiment can result in exactly 7 heads, you can use the concept of combinations. The number of combinations of 10 items taken 7 at a time is denoted as \( \binom{10}{7} \).
- The formula for combinations is given by:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]
where \( n \) is the total number of trials (10 coin tosses) and \( k \) is the number of successful trials (7 heads).
2. **Part (b): Sample Space Calculation**
- The sample space for tossing a coin 10 times includes all possible sequences of heads (H) and tails (T). Each toss has 2 possible outcomes, and for 10 tosses, the total number of outcomes is:
\[
2^{10} = 1024
\]
### Answers
a) There are \( \binom{10}{7} \) ways for the experiment to result in exactly 7 heads.
b) The size of the sample space for the experiment is \( 2^{10} = 1024 \) possible outcomes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ae4e595-d9b4-4096-a3fc-f4bee7947b38%2Fa2b6902e-18e0-423e-b327-b09a1298876a%2F1rhn6g_processed.png&w=3840&q=75)
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