Probability 5: Binomial Distribution Pearson: Chapter 15.2 p534 - 540 Homework: Exercise 15.2 Q2, 3, 6 - 12 Haese & Harris: Chapter 23 p618 - 626 Homework: Ex 23D.2 evens Ex 23D.3 Q 2-4

Binomial Distribution Binomial Distribution (of a discrete variable X) Represents the distribution of the number of successful outcomes of a binomial experiment (such as tossing a coin to get “heads”), where there is a known number of trials (how many tosses) and the probability of success in each trial is known (p(heads) = 0.5 for a fair coin). So if the coin is tossed 3 times, the binomial distribution gives the probability of getting 0, 1, 2 or 3 heads.

Binomial Distribution X ~ B(n,p) = random variable X is distributed binomially with “n” number of trials, and a probability of success “p”. Example 1. If X ~ B(5, 0.6); find P(X = 4). (Find the probability that out of 5 trials, we get exactly 4 successes, where p = 0.6)

Binomial Distribution with GDC Use Binomial pdf (probability distribution function) for finding the probability of an exact value. Use Binomial cdf (cumulative distribution function) for finding the probability of all the values less than or equal to a value. 2. a) If X ~ B(4, 0.25); find P(X = 3). TI 84: 2nd VARS (dist); A: binompdf ENTER; trials: p: x value: ENTER; correct notation to write for your answer, ENTER b) If X ~ B(4, 0.25); find P(X ≤ 2). TI 84: 2nd VARS (dist); B: binomcdf ENTER; trials: p: x value: ENTER; correct notation to write for your answer, ENTER c) If X ~ B(4, 0.25); find P(X ≥ 3). The Ti84 can’t add probabilities more than a value so this can be thought of as P(X ≥ 3) = 1 – P(X ≤ 2)

Binomial Distribution Example 3. Records show that 6% of the items assembled on a production line are faulty. A random sample of 12 items is selected at random (with replacement). Find the probability that: None will be faulty b) at most, one will be faulty c) At least two will be faulty d) at least one is faulty

Binomial Distribution Example 4. There are 2 blue marbles in a bag of 10 marbles. If a marble is selected, its colour noted, and then returned to the bag, how many trials does it take so that the probability of getting at least 1 blue marble is greater than 0.95?

Expected Value of a Binomial Distribution The expected value or mean of a random variable X is the average value we would expect for X over many trials of the experiment. Notation = E(X) Example 5. On the day of a math test, the probability of a student being absent is 11%. If 73 students sit the next math test, how many students can be expected to be absent?

Binomial Distribution and Expected Value Example 6. A tennis player finds that he wins 3 out of 7 games he plays. If he plays 7 games straight, find the probability that he will win: exactly 3 games at most 3 games all 7 games more than 4 games at least 1 game f) after playing 30 games, how many of these would he expect to win?