`25:00`

STAT 20: Introduction to Probability and Statistics

- PS 8: time to work on it and then review
- Brief lecture and CQs on Random Variables
- Break
- PS 9 (random variables)
- If time, coin flipping graph

`25:00`

Let \(X\) be the number of heads in three tosses of a fair coin.

What about if \(X\) is the number of heads in 3 tosses of a biased coin, where the chance of heads is \(\frac{2}{3}\)?

Now suppose we toss a fair coin until the first time it lands heads, and let \(X\) be the number of tosses. What is the pmf of \(X\)? Is it binomial?

Finally, let’s consider a deck of cards, and we are interested in the number of hearts dealt in a hand of five. Call this number \(X\). What is the pmf of \(X\)?

\(f(x)\) is the probability mass function of \(X\). What does that mean? What is the connection to the distribution table? The probability histogram?

\(F(x)\) is the cumulative distribution function.

What is the connection between \(f\) and \(F\)?

`01:00`

Is the following random variable binomial (if so, what are \(n\) and \(p\)?), hypergeometric (if so , what are \(N\), \(G\), and \(n\)?), or neither?

Roll a fair ten-sided die 20 times. Let \(X\) be the number of times we roll a multiple of 3.

Binomial, hypergeometric, or neither?

`01:00`

Is the following random variable binomial (if so, what are \(n\) and \(p\)?), hypergeometric (if so , what are \(N\), \(G\), and \(n\)?), or neither?

YouGov surveyed about 1,900 adults and asked them if they thought that Kevin Mcarthy should be ousted from his role as speaker. Let \(X\) be the number of people who responded “Yes”. The population of the US is about 335 million.

Binomial, hypergeometric, or neither?

`01:00`

Is the following random variable binomial (if so, what are \(n\) and \(p\)?), hypergeometric (if so , what are \(N\), \(G\), and \(n\)?), or neither?

A six-sided die is tossed two times and the sum of the faces showing is \(8\). Let \(X\) be 1 if the sum is \(8\) and \(0\) otherwise.

Binomial, hypergeometric, or neither?

`01:00`

A bag that has 6 pieces of fruit: 2 mangoes, 3 apples, and 1 orange. I reach into the bag and draw out one fruit at a time, selecting each fruit at random (so they are equally likely). Let \(X\) be the number of draws until and including the first time I draw a apple.

Binomial, hypergeometric, or neither?

`01:00`

You have \(10\) people with a cold and you have a remedy with a \(20\%\) chance of success. What is the chance that your remedy will cure at least *one* sufferer? (Let \(X\) be the number of people cured among the 10. We are looking for the probability that \(X \ge 1\))

What is the chance that at least one person is cured?

`03:00`

Roll a pair of fair six-sided dice and let \(X = 1\) if the dice land showing the same number of spots, and \(0\) otherwise. For example, if both dice land \(2\), then \(X = 1\), but if one lands \(2\) and the other lands \(3\), then \(X = 0\).

What is \(P(X=1)\)?

`05:00`