Math  /  Data & Statistics

QuestionLa dyschromatopsie, trouble de la vision des couleurs le plus courant (également appelé daltonisme) affecte, en France 8\% des hommes et 0.4%0.4 \% des femmes. A l'aide d'une approximation pertinente, calculer la probabilité que, sur 500 femmes, une au moins présente ce trouble.

Studdy Solution

STEP 1

What is this asking? What's the chance that at least one woman out of 500 has color blindness? Watch out! It's *much* easier to figure out the chance that *no one* has it, and work from there.

STEP 2

1. Probability of *not* having color blindness
2. Probability of at least one person having color blindness

STEP 3

We're given that 0.4%0.4\% of women have color blindness.
This means the probability of a randomly selected woman having this condition is 0.4%=0.4100=0.0040.4\% = \frac{0.4}{100} = 0.004.

STEP 4

Now, if the probability of having color blindness is 0.0040.004, the probability of *not* having it is 10.004=0.9961 - 0.004 = 0.996.
Remember, the total probability of all possibilities must add to one!

STEP 5

We want to know the probability that *none* of the **500** women have color blindness.
Since each woman is independent of the others, we can simply multiply the individual probabilities together.
So, the probability that *none* of the 500 women have color blindness is (0.996)500(0.996)^{500}.

STEP 6

Let's calculate that: (0.996)5000.135(0.996)^{500} \approx 0.135.
So, there's roughly a **13.5%** chance that *none* of the women have color blindness.

STEP 7

We just figured out that there's a 0.1350.135 probability that *no one* has color blindness.
We want the probability that *at least one* person does.
These two scenarios are the only possibilities, and their probabilities *must* add up to **one**!

STEP 8

Therefore, the probability that at least one woman has color blindness is 10.135=0.8651 - 0.135 = 0.865.

STEP 9

There's approximately an **86.5%** chance that at least one of the 500 women has color blindness.

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