14.5 - Probability

Links and useful resources

• gr7's Prealgebra Course Outline START HERE<<<
• AoPS Online Textbook
• IXL Grade 7 index
• IXL Grade 8 index
• AoPS Alcumus
• IXL prealgebra practice index
• OpenSTAX Prealgebra-1 textbook

Concept summary and lesson

• Probability is the fraction of outcomes you care about vs all possible outcomes that matter.
• Probability doesn't tell us exactly what will happen! It tells us how likely different things are to happen if we try something over and over. If you think of the different possibilities as buckets, and the trials as "throwing a grain of sand into one of the buckets," then probability tells us approximately how much each bucket will weigh after we do millions of trials. It doesn't tell us the exact number of sand grains, but it does give a very good estimate of the overall situation at the end.

First off, we have to make a new term: discrete event. A discrete event is something that's countable with whole-numbers, like "how many times a die rolls 3." There are other things that really don't make sense to count that way, like "how much water is in the container?" We're going to be working with discrete things for now.

Computing probabilities for discrete events generally comes down to a short process that looks like this:

1. Figure out how to count the number of possible ways something could turn out
2. Figure out how to count the number of ways where it turns out being what we're wanting
3. Divide the second by the first.

The counting is the hard part!

Guided practice

• The faces of a fair 6-sided die are numbered 1-6. (“Fair” means that each side is equally likely to be rolled.) What is the probability that when the die is rolled, a 2 is facing up?
• What is the probability that when a fair 6-sided die is rolled, an odd number faces up?
• What is the greatest possible probability for any event? What is the least possible probability for any event?
• A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\mathrm{♡}$ and $\mathrm{♢}$, called “hearts” and “diamonds”) are red, and the other two ($\mathrm{♠}$ and $\mathrm{♣}$, called “spades” and “clubs”) are black. The cards in the deck are placed in random order (usually by a process called “shuffling”). What is the probability that the top two cards are both red (this is tricky!)?
• Suppose we have 2 fair 6-sided dice numbered from dice-blue-1 to dice-blue-6. If we roll them both, then what is the probability that the two numbers shown sum to 7?

Homework

• #hw (math) (gr7) 14.5 Probability (p. 536) 2025-05-02