A coin is tossed twice. Let Z denote the number of heads on...

A coin is tossed twice. Let $Z$ denote the number of heads on the first toss and $W$, the total number of heads on the 2 tosses. If the coin is unbalanced and a head has a $30 \%$ chance of occurring, find (a) the joint probability distribution of $W$ and $Z$; (b) the marginal distribution of $W$; (c) the marginal distribution of $Z$; (d) the probability that at least 1 head occurs.

A coin is tossed twice. Let $Z$ denote the number of heads on the first toss and $W$, the total number of heads on the 2 tosses. If the coin is unbalanced and a head has a $30 \%$ chance of occurring, find
(a) the joint probability distribution of $W$ and $Z$;
(b) the marginal distribution of $W$;
(c) the marginal distribution of $Z$;
(d) the probability that at least 1 head occurs.

Key Concepts

Complement Rule

The complement rule is a fundamental concept in probability which states that the probability of an event occurring is one minus the probability of its complement (the event not occurring). This rule simplifies calculations, especially when directly computing the desired probability is complex. For instance, finding the probability of at least one occurrence of an event can be more easily determined by subtracting the probability of no occurrences from one.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is applicable to scenarios such as coin tosses, where each toss is independent and has two possible outcomes. Understanding this distribution helps in calculating the probabilities associated with sequences of independent events.

Joint Probability Distribution

The joint probability distribution describes the likelihood that two random variables simultaneously take on a given pair of values. In problems where the outcomes of multiple experiments or trials are considered (such as coin tosses), the joint distribution assigns probabilities to all possible combinations of outcomes. This concept is crucial when analyzing how the variables interact, and when computing probabilities of events that involve both variables.

Marginal Probability Distribution

The marginal probability distribution of a random variable is obtained by summing or integrating the joint probability distribution over the other variable(s). In the context of multivariate distributions, it provides the probabilities of the outcomes of one variable irrespective of the others. This is particularly useful when one is interested in the behavior of a single variable in a multivariable framework.

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