Cleerelearn

Category: Probability

  • Why Do You Multiply Probability by the Total?

    You’re in the middle of a probability question and it says something like:

    The probability of rolling a 6 is 0.3.
    The dice is rolled 200 times.
    Estimate how many times it lands on 6.

    And everyone (including your teacher) immediately does this:

    0.3 × 200

    …and somehow gets the answer.

    But if you’re thinking:

    “Why are we multiplying? Where did that come from?”

    Good. That question is the difference between copying a method and actually understanding probability.

    Let’s make it make sense.

    The idea: probability is “out of 1”, frequency is “out of lots”

    A probability is basically saying:

    “Out of 1 attempt, what fraction of the time would this happen?”

    So if the probability is 0.3, that means:

    “This should happen about 30% of the time.”

    Now if you don’t do it once…
    You do it 200 times…

    You’re asking a different question:

    “If it happens 30% of the time, how many times is that out of 200?”

    That’s why we multiply.

    A simpler example first: coins

    If you flip a fair coin:

    • Probability of tails = 0.5
    • If you flip it 200 times, how many tails should you expect?

    Half of 200 is 100.

    So you do:

    0.5 × 200 = 100

    That multiplication isn’t random.

    The rule (this is the whole topic)

    If something has probability pp
    and you try it nn times,

    then the estimated number of times it happens is:

    expected frequency (or relative frequency) = probability × total trials

    Back to the dice question

    Probability of a 6 = 0.3
    Number of rolls = 200

    So the estimate is:

    0.3 × 200 = 60

    Meaning:

    If you rolled this biased dice 200 times, you’d expect about 60 sixes.

    Not guaranteed. Just the best estimate.

    Why this works (the “feel” for it)

    Think of it like this:

    • Probability tells you the “rate”
    • Total tells you “how many chances you have”
    • Multiply them to get “how many hits you expect”

    It’s the same logic as:

    “If 30% of the class is left-handed, and there are 200 students… how many left-handed students would you expect?”

    Same exact maths.

    Quick check you can do in your head

    Before you even calculate, do this:

    • 0.3 is about a third
    • a third of 200 is about 66
    • so the answer should be somewhere around 60–70

    So 60 makes sense.

    This is how you stop silly mistakes.

    Final thought

    You multiply probability by the total because:

    Probability is a proportion.
    The total tells you how many chances you have.
    Multiplying applies the proportion to the total.

    That’s all relative frequency really is:
    turning probability into an estimate.