Independent Events

Events are considered independent events if the occurrence of one has no bearing on the probability of the occurrence of the other. A good example to illustrate this is the tossing of a fair coin. A coin has no memory, meaning that even if a head comes up 10 times in a row, the probability that the next toss is a head is still ½. The probability that the next toss is a tail is also ½. The probability that the toss of a fair coin lands heads up or tails up on any single toss is always ½ no matter how many times the coin has been previously tossed.

If two events A and B are independent, then P(A ∩ B) = P(A) ∙ P(B).

• Note that the rule for independent events can be extended for more than 2 events. If there are 3 independent events, A, B and C, then P(A ∩ B ∩ C) = P(A) ∙ P(B) ∙ P(C). Similar results are seen for 4 events, 5 events and so on.

Example: Suppose a spinner has 12 regions numbered 1 – 12 each of which the spinner is equally likely to land on. What is the probability that the spinner will land on 2, 2, 10, 9 and any odd number in 5 consecutive spins?

P(2) = 1/12

P(10) = 1/12

P(9) = 1/12

P(odd) = ½

Therefore, since these are independent events, the probability the spinner lands on 2, 2, 10, 9 and any odd number in 5 consecutive spins is

(1/12)(1/12)(1/12)(1/12)(1/2) = 1/41,472

When factoring a binomial such as

4x^2 - 81, look to see if either of the terms are perfect squares. If they are, factoring is very simple.

Take the square root of each term to get

2x and 9

(notice we ignore the - on the 81)

The factored form is just two factors with each term, one with a + in between and one with a - in between

(2x + 9)(2x - 9)

If the terms of a binomial are not both perfect squares, then look for a greatest common factor to factor out.

Did you know that a rational function can also represent inverse variation?

For example, a rational function is a polynomial divided by another polynomial.

y = 20/x is a polynomial function, but it's also inverse variaton.

y = k/x reprsents inverse variation, y varies inversely as x and k is the constant.

In the above example, the constant is 20.

Consider these two problems involving volume.

The diameter of a baseball is approximately 7.37 cm. A hockey puck is cylindrical with a thickness

of 2.54 cm and a diameter of 7.6 cm. Which has the greatest volume?

Solution:

A baseball is spherical and the volume of a sphere is (4/3)πr³. The radius of the sphere is half the diameter, so

r = 3.685 cm. Using 3.14 for π, the volume of the baseball is (4/3)(3.14)(3.685)³ = 209.5 cm³. (rounded to one decimal place)

The volume of a cylinder is πr²h. The radius of the hockey puck is 3.8 cm, the height is 2.54 cm. Therefore the volume of the hockey puck is (3.14)(3.8)²(2.54) = 115.2 cm³. (rounded to one decimal place)

The volume of a baseball is nearly double the volume of a hockey puck.

Suppose you have a metal cone shaped container and a plastic container in the shape of a shoe box. The cone shaped container is 14 inches high with a diameter of 10 inches. The plastic container is 12 inches long, 6 inches wide and 4 inches high. You want to fill up one container with water. Which container will hold the most water?

Solution:

The volume of a cone is (1/3) πr²h.

Therefore, the volume of the cone shaped container is (1/3)(3.14)(5)²(14) = 366.3 inches³ (rounded to one decimal place)

The volume of a rectangular solid is length times width times height, therefore the volume of the plastic container is (12)(6)(4) = 288 inches³.

The metal cone shaped container will hold the most water.

What is 2 X 2 + 2 X 2 + 2 - 2 X 2 ?

Be careful with a problem like this. People might say there are several answers depending on what operation is performed in what order. That's why there are the order of operations to follow.

By following the order of operations, the answer is 6. Follow the order of operations and perform from left to right. Multiplication comes before addition and addition comes before subtraction.

4 + 4 + 2 - 4
10 - 4
6

The time it takes a home building company to build a new home is directly proportional to the size of the home and inversely proportional to the number of people on the building crew. If it takes 10 workers 4 weeks to build a 2,000 square foot home, how long will it take 6 workers to build the home?

Solution:

Let t = time it takes to build a home

w = number of workers

s = size of home

The joint variation between these variables can be represented by

t = ks/w

Substitute 4 for t, 2000 for s and 10 for w and solve for k.

4 = k(2000)/10

Multiply both sides by 10 to clear the denominator.

40 = k(2000)

Divide by 2000 to get

40/2000 = 1/50 = k.

Now substitute 6 for w, 2000 for s and 1/50 for k and solve for t.

t = (1/50)(2000/6)

t = 2000/300, t = 6.66. Therefore, it takes 6 workers 6 2/3 weeks to build a 2,000 square foot home.