Coin ToSS Lab n this lab, we will examine the Law of Large Numbers and use it to determine if a coin is unfair. A fair coin is equally weighted on each side. An unf s a coin that weighs more on one side so that the chances of it landing on a certain side is more likely. n this activity, $$ H $$ will represent the coin landing heads up and $$ T $$ will represent the coin landing tails up. 1. Before you toss the coin, use theoretical probability to determine the probability of the coin landing heads up and the probability of the coin la heads down. $$ P(H)= $$ (Click to view hint) $$ P(T)= $$ (Click to view hint)

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### Step 1: Theoretical Probability Before Tossing the Coin

Before performing any coin tosses, it's important to establish the theoretical probabilities based on whether the coin is **fair** or **unfair**.

#### **Fair Coin**
A **fair coin** has no bias, meaning both sides are equally likely to land face up.

- **Probability of Heads ($$ P(H) $$):**
  $$
  P(H) = \frac{1}{2} = 0.5 \text{ or } 50\%
  $$
  
- **Probability of Tails ($$ P(T) $$):**
  $$
  P(T) = \frac{1}{2} = 0.5 \text{ or } 50\%
  $$

#### **Unfair Coin**
An **unfair coin** is biased, meaning one side is more likely to land face up than the other. Let’s denote the probability of landing heads as $$ P(H) $$ and tails as $$ P(T) $$.

- **Probability of Heads ($$ P(H) $$):**
  $$
  P(H) = p \quad \text{where} \quad 0 < p < 1
  $$
  
- **Probability of Tails ($$ P(T) $$):**
  $$
  P(T) = 1 - p
  $$

*Note:* The value of $$ p $$ would depend on how biased the coin is. For example, if the coin is weighted to land heads 70% of the time:
$$
P(H) = 0.7 \quad \text{and} \quad P(T) = 0.3
$$

### Summary

- **Fair Coin:**
  $$
  P(H) = 0.5 \quad \text{and} \quad P(T) = 0.5
  $$
  
- **Unfair Coin:**
  $$
  P(H) = p \quad \text{and} \quad P(T) = 1 - p \quad \text{where} \quad p \neq 0.5
  $$

Use these theoretical probabilities as a baseline for your experiments. After performing multiple coin tosses, you can compare the experimental probabilities with these theoretical values to determine if the coin is fair or unfair using the Law of Large Numbers.
Before tossing the coin, the probability of it landing heads up ($$ P(H) $$) is 0.5, and the probability of it landing tails up ($$ P(T) $$) is also 0.5.

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