Probability: Bayes, counting and Bernoulli trials · Grade 11 · 5 units
In Probability: Bayes, counting and Bernoulli trials (Grade 11 · 5 units) you practise 7 skills: Probability from areas and lengths · Bayes’ rule · The multiplication principle · Arrangements and roles · Choosing without order · Bernoulli trials · At least and at most: the complement. In this topic probability is built from counting. When every outcome is equally likely, the probability is the number of outcomes that fit over the number of all outcomes. To count, use the multiplication principle: choose in steps, and multiply the numbers of options at each step. When the order matters, count arrangements; when it does not, divide by the number of orders of each group.
When the same trial is repeated several times and the trials are independent, the probability of a run of results is the product of the probabilities. The probability of exactly a given number of successes is the number of runs that fit times the probability of one run: these are Bernoulli trials. For at least once, it is easier to find the complement. Bayes’ rule turns a question around. When a medical test comes out positive, the question is how likely it is that the person is really ill. A simple way is to think of a hundred thousand people, count each group in a table, and divide the ill people with a positive result by all the positive results. When the illness is rare, the answer is smaller than expected.
Tip: Before counting, ask whether the order matters. If it does, count arrangements; if not, divide by the number of orders.
What you learn
Probability from areas and lengths
For example: A dart lands at random on a square board with side 2. A circle with radius 1 is drawn on the board, touching every side. What is the probability that the dart lands in the circle?
Answer and solution
The answer: π/4
The area of the circle is π , and the area of the square is 4 . The probability is π ÷ 4 , that is π/4 .
Bayes’ rule
For example: A certain illness affects 0.1 percent of the population. A test finds 90 percent of the people who are ill: their result is positive. For 96 percent of the healthy the result is negative. A person is picked at random and tested. What is the probability that the result is positive? Give the answer as a percentage, to 1 decimal place.
Answer and solution
The answer: 4.1%
Out of 100,000 people 100 are ill, and 90 of them test positive. Of the 99,900 healthy, 3,996 test positive. That makes 4,086 positive results out of 100,000, about 4.1 percent.
The multiplication principle
For example: A restaurant has 2 starters, 5 main courses and 2 desserts. A meal is one starter, one main course and one dessert. How many different meals can be made?
Answer and solution
The answer: 20
By the multiplication principle: 2 × 5 × 2 = 20 .
Arrangements and roles
For example: A class has 5 pupils. Four pupils are chosen for four different roles, one pupil for each role. In how many ways can this be done?
Answer and solution
The answer: 120
The first role has 5 options, the second 4, and so on: 5 × 4 × 3 × 2 = 120 .
Choosing without order
For example: In a group of 4 friends, every two of them shake hands once. How many handshakes are there?
Answer and solution
The answer: 6
Each handshake is a choice of two from 4, with no order: 4 × 3 ÷ 2 = 6 .
Bernoulli trials
For example: A spinner has 4 equal parts, and one of them is red. It is spun twice. What is the probability that it never lands on red?
Answer and solution
The answer: 9/16
On each spin the probability of not landing on red is 3/4 . So the probability is (3/4)² , that is 9/16 .
At least and at most: the complement
For example: A spinner has 6 equal parts, and one of them is red. It is spun twice. What is the probability that it lands on red at least once?
Answer and solution
The answer: 11/36
The complement is that it never lands on red: (5/6)² = 25/36 . So the probability is 1 − 25/36 , that is 11/36 .
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).