When dropped on a hard surface a thumbtack lands with its...

When dropped on a hard surface a thumbtack lands with its sharp point touching the surface with probability $2 / 3 ;$ it lands with its sharp point directed up into the air with probability $1 / 3$. The tack is dropped and its landing position observed 15 times. a. Find the probability that it lands with its point in the air at least 7 times. b. If the experiment of dropping the tack 15 times is done repeatedly, what is the average number of times it lands with its point in the air?

When dropped on a hard surface a thumbtack lands with its sharp point touching the surface with probability $2 / 3 ;$ it lands with its sharp point directed up into the air with probability $1 / 3$. The tack is dropped and its landing position observed 15 times.
a. Find the probability that it lands with its point in the air at least 7 times.
b. If the experiment of dropping the tack 15 times is done repeatedly, what is the average number of times it lands with its point in the air?

Key Concepts

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a given value. In the context of the binomial distribution, it often requires summing up individual probabilities of obtaining various counts of successes to find the total probability for a range of outcomes.

Bernoulli Trial

A Bernoulli trial is a random experiment where there are exactly two possible outcomes, commonly labeled as success and failure, with a constant probability for success in each trial. This concept is fundamental when modeling situations where each independent trial follows the same probability law.

Binomial Distribution

The binomial distribution is used to model the number of successes in a fixed number of independent Bernoulli trials. It encapsulates the probability of obtaining a specific number of successes, and its probability mass function involves the binomial coefficient and the probabilities of success and failure raised to appropriate powers.

Expected Value

The expected value of a random variable is a measure of the center of its distribution, calculated as the weighted average of all possible values. For a binomial distribution, the expected value is determined by multiplying the number of trials by the probability of success per trial, providing an average count of successes over repeated experiments.

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