Ratios, Proportions and Probability
A ratio is a way of showing a relationship between two values. For example, if the grocery store was selling soup at a rate of 5 cans for $3.00, the ratio of cans to dollars would be 5 to 3. This ratio can be written a few ways: written out as 5 to 3, with a colon as 5:3 or, most commonly, as a fraction 5/3.
Probability is the likelihood that something will happen, and it is usually shown as a ratio. If you have a bag of marbles, and there are 4 red ones, 2 blue ones, and 5 yellow ones, you can calculate the probability that you could reach in and draw a red marble. There are 11 total marbles in the bag and 4 of them are red, so your probability is 4 in 11, or 4/11.
Combinations and Permutations
Sometimes in probability you may have to think how likely something is to happen more than one time. For this, we use combinations. A combination involves multiplying something to increase its possibilities. For example, if a restaurant offered a deal on a sandwich with a side and a drink for a set price, they might advertise the number of combinations you could make for a meal. If there are 5 sandwiches, 3 sides and 6 possible drinks, you find the number of combinations by multiplying 5 • 3 • 6 = 90. Likewise, if you have the above bag of marbles and you want to know your chances of drawing a red marble twice in a row, you multiply your probability times itself. Your probability of drawing a red marble each time is 4/11, so your probability of drawing it twice is 4/11 • 4/11 = 16/121. That is, of course assuming you threw your first marble back and had 11 marbles in the bag for each draw. If not, we need a permutation.
With a permutation, your probability changes with each repetition. If your probability of drawing a red marble is 4/11 on your first draw, if you succeed, the number of marbles and red marbles in the bag has changed. On your second draw there will be 10 marbles in the bag and only 3 of them will be red. Now, your probability of drawing a red marble will be 4/11 on your first draw and 3/10 on your second draw, which combines to 4/11 • 3/10 = 12/110 which reduces to 6/55. 6/55 is your probability of drawing two red marbles.
When you have two ratios that are equal, they are called proportionate. For example, if the store was selling soup cans at a rate of 5 for $3.00, you could also say that they were selling 20 cans for $12.00. The proportion you could write to express this would be 5/3 = 20/12
Using Proportion to Solve Problems
If you have a missing value in an equal set of ratios, you can use a proportion to solve them. Suppose I told you that I was going on a trip and driving a constant 60 miles per hour. Then, I asked how long it would take me to drive 300 miles. Let’s set up a ratio for my speed: 60 miles per 1 hour, or 60/1. Then an equivalent ratio for how far I have to drive: 300 miles in an unknown amount of time, or 300/x. Now, we can set these up as a proportion: 60/1 = 300/x. Since we know that equivalent fractions can be identified by cross-multiplying, we know that 60 times the missing value will equal 1 times 300. Therefore, we can solve any proportion by multiplying the values diagonal from each other, then dividing by the other number. 1 × 300 = 300; 300 ÷ 60 = 5. The trip will take us 5 hours.