A coin is biased so that tails is twice as likely to occur as heads. If the coin is tossed 4 tim

A coin is biased so that tails is twice as likely to occur...

Key Concepts

Biased Probability

This concept refers to situations in which not all outcomes are equally likely. In probability theory, a biased event means that the probability of success and failure are not equal, and the bias must be accounted for when calculating outcomes. This impacts calculations in experiments where events are not uniform, thereby affecting the overall probability distribution.

Bernoulli Trials

A Bernoulli trial is an experiment that results in exactly two outcomes, often denoted as "success" and "failure." Each trial is independent and has its own probability of success. When an experiment is repeated multiple times under identical conditions, each outcome can be modeled as a Bernoulli trial, making it fundamental to understanding experiments involving binary outcomes.

Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials. It is used to model experiments where each trial has two outcomes. The probability of obtaining a specific number of successes is given by the binomial formula, which incorporates the probability of success, the probability of failure, and the number of ways the successes can occur.

Combinatorial Analysis

This concept involves calculating the number of ways an event can occur, typically using combinations. In probability problems, combinatorial analysis is used to determine how many different arrangements or selections meet a certain criterion, such as obtaining a specific number of successes in a series of trials. It is essential for determining the number of favorable outcomes.

Cumulative Probability

Cumulative probability refers to the probability of obtaining a range of outcomes, usually expressed as 'at least' or 'at most' a certain number of successes. It is calculated by summing the individual probabilities of each outcome within the defined range. This concept is especially important when a problem requires the probability of multiple outcomes rather than a single event.

Transcript

00:03 So we have an unfair coin, a biased coin, and tails is twice as likely to occur as heads.

00:10 And that is what the second equation represents, that scales is twice as likely to occur as heads.

00:17 And our first equation is this basic foundation of statistics and that the sum of the outcomes has to equal one, right? the probability of heads plus probability of tails has equal one.

00:28 In a fair coin, it would be 50 -50, but this coin is unbiased.

00:32 So, or excuse me, it is biased.

00:34 So our probability is going to change a little bit.

00:37 And how we can figure that out is using substitution.

00:41 So we can sub the probability of t in our first equation with two times the probability of h because the second equation says that they are equal.

00:52 So when we rewrite that, we have two times the probability of h plus the probability of h is going to be equal to 1.

01:09 And now we have two times probability of h plus the probability of h itself, and that gives us three times the probability of h because we can treat the probability as a variable or as if we are combining like terms.

01:26 So we have three times the probability of heads equals 1.

01:31 So when we solve for the probability of heads, we divide by three and we get that the probability of heads is one -third.

01:41 And going back to our first equation, where the sum has to equal one, we know that the probability of tails has to be two -thirds.

01:54 So knowing that, we can get on to solving our questions.

01:57 So the first one wants to know what the probability of getting exactly two heads is if we toss a coin five, this coin four times.

02:06 So, i listed all the ways that you can get exactly two heads.

02:10 There are six of them.

02:12 Now, if say if we were to take this, just this first scenario, we get two heads in a row and then two tails.

02:18 We would get one third times one third times two thirds times two thirds, right? we have the probability of heads and then heads again and then tails and the tails.

02:32 But if we look at the other five ways to get exactly two heads, the values, the probabilities are the same, they might just be a different order.

02:41 For example, this next one, our tails, tails, heads heads would be two thirds times two thirds, times one thirds...

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