Lesson 3 of 4

Or, and, not.

About 25 minutes · 5 practice questions

How events combine. At least one six with two dice through the complement, why you cannot add the chances of events that can happen together, when you multiply, and why a coin has no memory.

Step 1 of 5 · At least one six

Step 1 of 5

At least one six

Roll two dice. What is the chance of at least one six? You could count it directly, but it is easier to count the complement: “neither die is a six”.

The board lights the squares with no six at all. The first die has $5$ ⁠ faces that are not a six, and so does the second, which makes $5 \times 5 = 25$ ⁠ squares. All the others are the squares with at least one six.

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Step 2 of 5

A six on the first or the second

Now two events on the same table: a six on the first die (sky blue) and a six on the second die (pink). The event “a six on the first or the second” takes in every square coloured by either one, and also the squares that belong to both, which are coloured amber.

Each event covers $6$ ⁠ squares. Predict the chance of “or”, then look at the corner of the table.

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Step 3 of 5

Events that cannot happen together

When no square belongs to both events, they are called mutually exclusive, and only then can you simply add: the chance of “ $A$ ⁠ or $B$ ⁠” is $P(A) + P(B)$ ⁠.

When they overlap, the overlap is counted twice, so you take away once the chance that both happen together. Here the first event is a total of $7$ ⁠. Change Second total so that the two events together cover exactly a quarter of the table.

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Step 4 of 5

Three heads

And works like the stages of a tree: each toss splits in two, and three tosses give $2 \times 2 \times 2 = 8$ ⁠ equally likely paths. When the tosses do not affect each other they are independent, and the chance that all of them succeed is the product of their chances.

Predict the chance of three heads, then watch the path light up.

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Step 5 of 5

A coin has no memory

The board shows four tosses, with the path of four heads lit: one path out of $16$ ⁠. Seen in advance, a run like that is rare.

But suppose it has already happened. The coin does not know what came before, and the next toss is a new toss, independent of the earlier ones.

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Interactive scene

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Each step sets up the Scene for you, and a control marked “fixed” stays put for that step.

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Practice · Calculate

$25$ ⁠ squares are lit: those with no six at all. In how many of the $36$ ⁠ outcomes is there at least one six?

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Practice · Predict, then look

A six on the first die: $\frac{6}{36}$ ⁠. A six on the second die: $\frac{6}{36}$ ⁠. What is the chance of a six on the first or the second?

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Practice · Reach the target

The first event is a total of $7$ ⁠ ( $6$ ⁠ squares). Change Second total so that “a total of $7$ ⁠ or the second total” covers exactly a quarter of the table, then press Check.

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Practice · Predict, then look

Toss a fair coin three times. What is the chance that all three tosses come up heads?

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Practice · Choose

A fair coin has landed heads four times in a row. What is the chance that the fifth toss comes up heads?