Probability & Statistics (11 of 62) The Probability Function - Flipping Coins - General Formula 2 By Michel van Biezen

Summary:

1. Introduction to General Equation: The lecturer introduces a more general equation for finding the probability of outcomes in situations involving binary events, such as flipping coins.

2. Parameters $�$ and $�$:

   • $�$ represents the total number of coins flipped, while $�$ represents the number of heads obtained.
   • $�$ can range from 0 to $�$, inclusive.

3. Sample Space and Event Probability:

   • The sample space consists of $2^{�}$ possible outcomes, where $�$ is the number of coins flipped.
   • The probability of an event, such as obtaining $�$ heads and $�-�$ tails, is given by $\frac{�\text{ choose }�}{2^{�}}$.

4. Definition of $�$ choose $�$:

   • It's defined as $\frac{�!}{�!\times (�-�)!}$, where $�!$ denotes the factorial of $�$.
   • Factorial is the product of all positive integers up to that number.

5. Example Calculation:

   • Using an example with 4 coins, they calculate the probability of getting 1 head and 3 tails.
   • Substituting $�=4$ and $�=1$ into the formula, they simplify to $\frac{4!}{1!\times 3!}÷16$.
   • After canceling out common factors in the factorial expressions, they find the probability to be $\frac{1}{4}$.

6. Conclusion:

   • The lecturer concludes by emphasizing that this method allows for the calculation of probabilities in situations involving multiple coin flips.
   • They note that the order of heads and tails is not considered in this calculation.

Overall, the lecture provides a step-by-step explanation of how to calculate the probability of specific outcomes in scenarios involving multiple coin flips, demonstrating the application of the general equation and factorial notation.