8. Worked Examples

Flipping a fair coin 5 times. 𝑛 = 5,𝑝 = 0.5. • Probability of exactly 3 heads:

5 𝑃(𝑋 = 3) = ( )(0.5)3(0.5)2 = 10×0.125×0.25 = 0.3125 3

In this example, we're flipping a fair coin 5 times. Each flip has a 50% chance of landing heads and a 50% chance of landing tails, which means our probability of success (getting heads) is 0.5. To find the probability of getting exactly 3 heads out of 5 flips, we can use the binomial probability formula. The formula involves calculating a combination of the number of successes (3 heads) from the total trials (5 flips) and then computing the probabilities of getting those heads and tails. This gives us a specific number—0.3125, which means there's a 31.25% chance of getting exactly 3 heads when flipping 5 coins.

Imagine you're playing a game where you flip a coin 5 times, and you're keeping score based on how many times you get heads. Knowing that you have about a 31% chance of scoring exactly 3 points can help you understand your likelihood of scoring in this game, making it more exciting and strategic as you play.

Same scenario. Probability of at most 2 heads: 𝑃(𝑋 ≤ 2) = 𝑃(0)+𝑃(1)+𝑃(2) Calculate each with formula or use calculator: final ≈ 0.5 (by symmetry).

Here, we are calculating the cumulative probability of getting at most 2 heads when flipping a coin 5 times. This means we want to find the total likelihood of getting 0, 1, or 2 heads. We can break it down into three separate calculations: the probability of getting 0 heads (all tails), 1 head, and 2 heads. By adding these probabilities together, we can find the total probability of having at most 2 heads. Interestingly, due to symmetry in flipping a fair coin, this total comes out to be approximately 0.5, indicating a balanced chance of getting 2 or fewer heads.

Think of it like a quiz where there are 5 multiple-choice questions, and you want to know how often you would get 0, 1, or 2 answers right by guessing. If the chances of guessing correctly lead to a 50% likelihood of getting at most 2 questions right, you would know that there’s a fair mistake zone you often land in while guessing.

With 𝑛 = 5,𝑝 = 0.5: • Mean = 5×0.5 = 2.5 • Variance = 5×0.5×0.5 = 1.25 • SD ≈ 1.118

In this example, we calculate the mean, variance, and standard deviation for our binomial distribution given that we have 5 trials (flips) and the probability of success (heads) is 0.5. The mean tells us the average number of heads we would expect to get, which is 2.5. Variance measures how much our results might differ from this average; with our calculations, it comes to 1.25. Standard deviation is simply the square root of variance and provides a measure of the spread of our data around the mean, resulting in about 1.118. These metrics help us understand the distribution of outcomes when flipping the coin several times.

Imagine this as tracking your performance over many games. If, on average, you score 2.5 points per game, knowing that you have a typical spread of about 1 point around that average informs you whether you're consistently performing or if results can be wildly different each time you play.

Rolling a die 10 times; success = “4 or less,” so 𝑝 = = .6667.

• Probability of at least 8 successes: 𝑃(𝑋 ≥ 8) = 1−[𝑃(0)+⋯+𝑃(7)]. Use calculator functions or tables.

In this scenario, we're rolling a die 10 times, and we define 'success' as rolling a number that is '4 or less.' This gives us a probability of success (p) of approximately 0.67. To find the probability of rolling 4 or less at least 8 times, we can use the complement rule, which states that the probability of at least 8 successes can be found by subtracting the probabilities of getting 0 through 7 successes from 1. This can involve multiple calculations, and it's often easier to use a calculator or probability tables to find these cumulative probabilities efficiently.

Think about a game where you're rolling a die repeatedly on your turn—each time hoping for a score that's 4 or lower. If you know you can usually align those scores around eight times in a row, that knowledge can spurt your confidence as you play, helping you strategize for your next moves with the odds in your favor.

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