Math Snap

STEP 1

What is this asking?
What's the probability of not A happening or B happening, if A and B are independent?
Watch out!
Don't mix up "and" and "or," and remember how independence works!

STEP 2

1. Find P(B)
2. Find P(not A)
3. Apply the formula for P(not A or B)

STEP 3

We're given $P(\bar{B}) = 0.6$, which means the probability of B not happening is 0.6.
Since B either happens or it doesn't, the probabilities must add to one!
So, $P(B) = 1 - P(\bar{B})$.

STEP 4

Let's calculate $P(B)$:
$$P(B) = 1 - 0.6 = 0.4$$ So, the probability of B happening is 0.4!

STEP 5

We're given $P(A) = 0.4$.
Similar to what we did before, since A either happens or it doesn't, we know $P(\bar{A}) = 1 - P(A)$.

STEP 6

Let's calculate $P(\bar{A})$:
$$P(\bar{A}) = 1 - 0.4 = 0.6$$ So, the probability of A not happening is 0.6!

STEP 7

Since A and B are independent, we can use the formula: $P(\bar{A} \cup B) = P(\bar{A}) + P(B) - P(\bar{A} \cap B)$, where $P(\bar{A} \cap B) = P(\bar{A}) \cdot P(B)$.
This formula tells us the probability of not A happening or B happening (or both!). The last term accounts for the overlap, so we don't double-count when both events happen.

STEP 8

First, let's calculate the probability of both not A and B happening:
$$P(\bar{A} \cap B) = P(\bar{A}) \cdot P(B) = 0.6 \cdot 0.4 = 0.24$$

STEP 9

Now, let's plug everything into our formula:
$$P(\bar{A} \cup B) = P(\bar{A}) + P(B) - P(\bar{A} \cap B) = 0.6 + 0.4 - 0.24$$

STEP 10

Finally, let's calculate the result:
$$P(\bar{A} \cup B) = 0.6 + 0.4 - 0.24 = 1 - 0.24 = 0.76$$

The probability of not A happening or B happening is 0.76!