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From 0 to 1, and placing an event on it before computing anything.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you place events on the scale from 0 to 1: impossible at one end, certain at the other, even chance in the middle. You compute probabilities by counting equally likely outcomes, and you find the chance that something does not happen. Getting a feel for the scale first is what stops an answer of 1.4 or $-0.2$ from going unnoticed.
You can write a fraction, change it to a decimal, and change a decimal to a percent: $\frac{1}{4} = 0.25 = 25\%$. You know how to compare fractions and decimals and place them on a number line between 0 and 1. You have also used words like likely, unlikely and fifty-fifty in everyday talk. This lesson puts numbers on those words.
| Term | What it means |
|---|---|
| Experiment | An action with an uncertain result, such as rolling a die or drawing a card. |
| Outcome | One possible result of an experiment. Rolling a die has 6 outcomes. |
| Event | A set of outcomes you are interested in, such as rolling an even number. |
| Probability | A number from 0 to 1 that tells how likely an event is. 0 is impossible and 1 is certain. |
| Favorable outcome | An outcome that makes the event happen. |
| Equally likely | Outcomes that each have the same chance, like the faces of a fair die. |
| Complement | The event that something does not happen. An event and its complement add to 1. |
A probability is a number that says how likely an event is. Every probability lies on a scale from 0 to 1.
When all the outcomes are equally likely, you can compute a probability by counting:
$$P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$$
The top can never be bigger than the bottom, and neither can be negative. That is why every probability lands between 0 and 1.
Another way: picture
Draw a line from 0 to 1 and mark 0.5 in the middle. Write impossible at 0, certain at 1 and even chance at 0.5. Every event you meet belongs somewhere on that line. Drawing a heart from a deck of cards goes at 0.25, a quarter of the way along.
Another way: steps
People describe chance with words, but words can be vague. The table links the common words with numbers.
| Word | Probability | Example |
|---|---|---|
| Impossible | 0 | Rolling a 9 on a six-sided die |
| Unlikely | between 0 and 0.5 | Rolling a 6: $\frac{1}{6} \approx 0.17$ |
| As likely as not | 0.5 | A fair coin landing tails |
| Likely | between 0.5 and 1 | Rolling more than 1: $\frac{5}{6} \approx 0.83$ |
| Certain | 1 | Rolling a whole number from 1 to 6 |
Two events can both be unlikely but still be very different. Rolling a 6 ($\approx 0.17$) is about 17 times more likely than a probability of 0.01. Numbers let you compare events that the words lump together.
A probability can be written in three ways, and they all mean the same thing. Drawing a red card from a standard 52-card deck has probability $\frac{26}{52} = \frac{1}{2} = 0.5 = 50\%$.
A weather forecast usually uses percents: a 30% chance of rain is a probability of 0.3. A board game rule might use a fraction: $\frac{1}{6}$ for rolling a particular number. A scientist might write a decimal like 0.002.
To place a fraction on the scale, turn it into a decimal by dividing. For $\frac{3}{8}$, $3 \div 8 = 0.375$, which is a little below the middle: unlikely, but not by much. Fractions are also easy to compare this way. Is $\frac{2}{5}$ or $\frac{3}{7}$ more likely? As decimals they are 0.4 and about 0.43, so $\frac{3}{7}$ is a little more likely.
The complement of an event is the event that it does not happen. If the event is rolling a 6, its complement is rolling a 1, 2, 3, 4 or 5.
An event and its complement cover every outcome, and they never happen together. So their probabilities always add to 1:
$$P(\text{not } A) = 1 - P(A)$$
This is often the quickest route. The probability of not rolling a 6 is $1 - \frac{1}{6} = \frac{5}{6}$. On the scale, an event and its complement sit the same distance from 0.5, one on each side: $\frac{1}{6}$ is as far below the middle as $\frac{5}{6}$ is above it.
The rule favorable over total only works when every outcome has the same chance. A spinner split into one half and two quarters has three sections, but landing on the big half is not $\frac{1}{3}$. It is $\frac{1}{2}$, because the half takes up half of the circle.
The fix is to split the spinner into equal parts first. Cut the half into two quarters, and the spinner has four equal parts, two of which are the old half. Now counting works: $\frac{2}{4} = \frac{1}{2}$.
Real life is full of outcomes that are not equally likely. A basketball player either makes a free throw or misses it, but a strong shooter makes about 8 out of 10, not 1 out of 2. A thumbtack dropped on a table lands point up or on its side, and those two outcomes are not equally likely either. For events like these you cannot count outcomes; you have to collect data and watch what happens, which is the subject of the next lesson.
Good problem solvers place an event on the scale before they work anything out. The estimate only needs to be rough: near 0, below the middle, about the middle, above the middle, or near 1.
Take a bag with 2 red and 9 yellow marbles. Before any arithmetic, you can see that red is the rare color, so picking red is unlikely, well below 0.5. When you compute $\frac{2}{11} \approx 0.18$, the answer agrees with your estimate, and you can trust it. If you had written $\frac{9}{11}$ by mistake, you would get about 0.82, far on the likely side, and the estimate would warn you straight away that something went wrong.
Estimating also helps when you compare two games. Suppose one game lets you win by rolling a 5 or a 6 on a die, and another lets you win by drawing a heart from a deck of cards. The first is $\frac{2}{6}$, about a third of the way along the scale. The second is $\frac{13}{52}$, exactly a quarter. Both are unlikely, but the die game gives you the better chance.
Every question in this lesson follows the same steps.
Each step has a reason. Estimating first means a wrong answer looks wrong. Checking that outcomes are equally likely makes sure counting is allowed. Counting the total carefully stops the most common error, which is dividing by the wrong number.
How to check. Any answer below 0 or above 1 is wrong, every time. If you get 1.4, you probably put the total on top. If the event is the more common one, the answer should be more than 0.5. Finally, find the complement a second way by counting the unfavorable outcomes: the two probabilities must add to exactly 1.
The National Weather Service gives a probability of precipitation for each forecast. A 40% chance of rain means there is a probability of 0.4 that at least 0.01 inch of rain falls at any given spot in the forecast area. On the scale, 0.4 is a little below the middle: rain is less likely than not, but not by much. The complement tells you the chance of staying dry: $1 - 0.4 = 0.6$. A family planning a picnic might go ahead at 0.4, but pack a tarp. At 90%, a probability of 0.9, the complement is only 0.1, and most people would move the picnic indoors.
In the Powerball lottery, a player picks five white balls from 69 and one red Powerball from 26. The chance of matching all six and winning the jackpot is 1 in 292,201,338. As a decimal that is about 0.0000000034, so close to 0 that it would not show on a scale drawn across a whole classroom wall. Compare it with rolling a 6 on a die, $\frac{1}{6} \approx 0.17$. Both events are unlikely, but the die event is about 50 million times more likely. A 2-dollar ticket buys a chance that is not zero, but is very near it.
A middle school gives each of the 250 families at its spring concert one ticket for a door prize, and draws 5 winning tickets. A family's chance of winning is $\frac{5}{250} = \frac{1}{50} = 0.02$, or 2%. The chance of not winning is $1 - 0.02 = 0.98$, almost certain. If the school draws 50 winning tickets instead, the chance becomes $\frac{50}{250} = 0.2$: still unlikely, but now one family in five goes home with a prize. Organizers use exactly this arithmetic to decide how many prizes to buy.
The most common mistake is dividing by the wrong total. With 3 red and 5 blue marbles, the probability of red is $\frac{3}{8}$, not $\frac{3}{5}$. The bottom counts every outcome, including the favorable ones.
The second is thinking that unlikely means impossible. An event with probability 0.05 happens about once in every 20 tries. Over many tries, unlikely events happen all the time.
The third is thinking every event with two outcomes is fifty-fifty. Either I win the raffle or I don't has two outcomes, but they are not equally likely.
The last is an answer off the scale. A probability of 1.25 or $-0.3$ is always a mistake. Check the answer lies between 0 and 1 before you move on.
Place the sun rises tomorrow morning.
$P = 1$
It happens every day, so it is certain.
Place rolling a 7 on a six-sided die.
$P = 0$
No face shows a 7, so it is impossible.
Place a fair coin lands heads.
$P = \tfrac{1}{2} = 0.5$
The coin has two equally likely sides, and one of them is heads.
Place a date picked at random from a week falls on a weekday.
$P = \tfrac{5}{7} \approx 0.71$
Five of the seven days are weekdays, which is more than half: likely.
Place a card drawn from a full deck is the ace of spades.
$P = \tfrac{1}{52} \approx 0.02$
Only one of 52 cards works, so it sits very close to 0: very unlikely.
A box holds 6 green, 10 blue and 8 red crayons. You pick one without looking. Count the favorable outcomes.
$\text{green} = 6$
Any green crayon makes the event happen.
Count all the outcomes.
$6 + 10 + 8 = 24$
Every crayon in the box is equally likely to be picked.
Write favorable over total.
$P(\text{green}) = \tfrac{6}{24}$
Equally likely outcomes let you count.
Simplify the fraction.
$\tfrac{6}{24} = \tfrac{1}{4}$
Divide the top and the bottom by 6.
Change it to a decimal and a percent.
$\tfrac{1}{4} = 0.25 = 25\%$
Decimals make it easy to place on the scale.
Place it on the scale.
$0 < 0.25 < 0.5 \Rightarrow \text{unlikely}$
One crayon in four is green, so most picks will not be green.
Cards numbered 1 to 20 are shuffled and one is drawn. Count all the outcomes.
$\text{total} = 20$
Each of the 20 cards is equally likely.
List the multiples of 3.
$3, 6, 9, 12, 15, 18 \Rightarrow 6 \text{ cards}$
These are the favorable outcomes for a multiple of 3.
Write favorable over total.
$P(\text{multiple of } 3) = \tfrac{6}{20}$
Favorable over total, with equally likely cards.
Change it to a decimal.
$\tfrac{6}{20} = \tfrac{3}{10} = 0.3$
Divide the top and bottom by 2, then read tenths.
Use the complement rule for not a multiple of 3.
$1 - 0.3 = 0.7$
An event and its complement add to 1.
Check the complement by counting.
$20 - 6 = 14, \quad \tfrac{14}{20} = 0.7$
Counting the other cards directly gives the same answer.
Place both on the scale.
$0.3 \text{ unlikely}, \quad 0.7 \text{ likely}, \quad 0.3 + 0.7 = 1$
They sit the same distance from 0.5, one on each side.
Count the favorable sections.
$\text{yellow} = 3$
Landing on any yellow section makes the event happen.
Count all the sections.
$\text{total} = 8$
The sections are equal, so each is equally likely.
Divide favorable by total.
Place it on the scale.
An event has a probability of $0.65$. Which word describes how likely it is?
A jar holds 20 marbles: $3$ red, $5$ blue and the rest green. One marble is picked without looking. Complete the worked solution.
Subtract the red and blue marbles from the total to count the green ones.
$20 - 3 - 5 =$ g
Every marble that is not red or blue is green.
Divide the green marbles by the total, as a decimal.
$P(\text{green}) =$ p
Each of the 20 marbles is equally likely, so probability is favorable over total.
Subtract from 1 to find the probability of not green.
$P(\text{not green}) =$ q
Green and not green cover every outcome, so they add to 1.
Only one of these numbers could be the probability of an event. Which one?
A bag holds $5$ red, $6$ green and $4$ blue marbles. You pick one without looking. What is the probability that it is red? Give a fraction or a decimal.
$P(\text{red}) =$ answer
A forecast gives a probability of $0.75$ that it rains in your town tomorrow. What is the probability that it does not rain?
$P(\text{no rain}) =$ answer
A fair six-sided die, numbered 1 to 6, is rolled once. How many faces show a number **less than** $1$, and what is the probability of rolling one? Give the probability as a fraction or a decimal.
Favorable faces: f. Probability: p
A school fair sells $200$ raffle tickets for one prize bike, and one ticket is drawn at random. You bought $10$ tickets. What is the probability that you win? Give a decimal.
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
For a class game, each letter of the state name MISSISSIPPI is written on its own tile, and the tiles go into a bag. A student draws one tile without looking. How many tiles are in the bag, how many show the letter P, and what is the probability of drawing one? Give the probability as a fraction or a decimal.
Tiles: n. Favorable: c. Probability: p
You can place an event on the probability scale. Without looking: what number means certain, what does a probability of 0.9 tell you about how often it happens, and if the chance of rain is 0.35, what is the chance of no rain?
18. Your turn: a spinner with 8 equal sections, 3 of them yellow, step 3
$\tfrac{3}{8} = 0.375$
Favorable over total, then divide for the decimal.
18. Your turn: a spinner with 8 equal sections, 3 of them yellow, step 4
$0.375 < 0.5 \Rightarrow \text{unlikely}$
Fewer than half the sections are yellow.