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346 PROBABILITY OF AN EVENT
Example 5. A coin is tossed and a die is rolled. What is the probability of obtaining a head and a prime number?
Solution. The result of tossing a coin cannot possibly affect the outcome of rolling a die. In other words, if the coin landed as a head, it would not affect the way the die would land. When there are two trials that have no effect on the outcome of each other, we say the outcomes are independent events.
The probability of tossing a head is \ and the probability of rolling a prime number with a die is f, because there are three numbers 2, 3, and 5 that are prime. When two events are independent we multiply the probabilities. Write
Probability of head and prime number = | x | Prob(head) x prob(prime)
Summary. If E\ and Ei are independent events, then
Probability of E\ and Ei = probability of E\ x probability of Ei
Experimental Probability The probability we have studied so far is theoretical probability, which is what we expect to happen based on a theoretical calculation. We now look briefly at experimental probability. This kind of probability is a prediction based on what has happened previously.
Example 6. Pat’s birthday is 29 April. What is the probability it will rain on Pat’s next birthday?
Solution. We need some historical data to make a prediction. Records for the past 10 years, say, are obtained from the Weather Office. We find that during the past 10 years it has rained three times on Pat’s birthday. Based on these data we predict what will happen this year:
Probability it will rain on Pat’s birthday = ^
A better result may be obtained if we go back 20 years or even longer.
Sometimes experimental results are compared with theoretical results. For example we know that if we toss a coin once, the probability of a head is |. Suppose we tossed a coin 50 times and recorded that 23 heads came up. We would say that the experimental probability of tossing a head is 23/50, which of course is not the same as |. However, the more times we toss the coin, the closer the experimental probability will be to the theoretical probability.
References: Canceling, Dice, Event, Odds, Prime Number, Tree Diagram.
Reference: Tree Diagram.
PRODUCT OF PRIME FACTORS
Reference: Cost Price.
Reference: Angle between a Line and a Plane.
To prove a statement is true is to present a logical argument that establishes the statement as a fact. The proof of a geometry theorem follows a logical argument that is often based on axioms or fundamental geometry theorems.
Example. Prove the theorem ‘The sum of the angles of a triangle = 180°.”
Proof. Suppose the three angles of the triangle are referred to as angles A, B, and C (see figure). We have to prove that A + B + C = 180°. In this proof it is necessary
to draw in an extra line through the point A and parallel to the line BC. Write
Angle x = angle B Angle y = angle C Angle x + angle A + angle y = 180° Therefore B + A + C = 180°
Alternate angles are equal Alternate angles are equal Sum of adjacent angles = 180° Since y = C and x = B
In proving this theorem, it was necessary to make use of two fundamental theorems that are found under the entries Adjacent Angles and Alternate Angles.
References: Adjacent Angles, Alternate Angles, Angles on the Same Arc, Geometry Theorems.
Two terms that are related to each other, but often confused, are ratio and direct proportion. You may wish to study the entry Ratio before continuing. When two ratios are equal we say they are in direct proportion. A good way of explaining direct proportion is to have a photograph enlarged. In the figure, the length of the smaller photograph is 3 cm and its width is 2 cm:
Canceling the fraction
In the enlarged photo, the length is 6 cm and the width is 4 cm:
_ , „ length 6
Ratio of = -
We say the lengths and widths of the two photographs are in direct proportion because the ratios of length/width for each photograph are equal.
If two figures are in direct proportion, we say they are similar. This means that two figures are similar if one figure is an enlargement of the other. Other quantities can be in direct proportion besides similar figures. For example, the extension of a spring is in direct proportion to the weight producing it, which is known as Hooke’s law.
Another kind of proportion is inverse proportion, which we now compare with direct proportion. An example of direct proportion is extending a spring by adding weights to it. As one quantity (the weight) increases, the other quantity (the extension of the spring) also increases in proportion. Inverse proportion also deals with two quantities, but as one quantity, increases the other quantity decreases.
Example. Suppose Harry has the job of painting a wall on both sides. He asks his friends for help, and five of them paint one side of the wall in 7 hours. Later, three more join him and the eight friends paint the other side of the wall. If all the painters work at the same rate, how long will it take them to paint the other side of the wall?