The probability that Ryan chooses Spanish is 0.86 and the probability that Riley chooses Spanish is 0.73. State the probability that:
a. they both chose Spanish
b neither of them choose Spanish
c. only one of them chooses Spanish
d. at least one of them chooses Spanish
a. If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
b. (1-.86)(1-.73) = ?
c. Either-or probabilities are found by adding the individual probabilities.
(1-.86) .73 = ?
OR
(1-.73) .86 = ?
d. This means one or both, so you need to add answers to a and c.
To find the probabilities requested, we can use basic probability rules.
a. To find the probability that both Ryan and Riley choose Spanish, we can multiply their individual probabilities:
P(both choose Spanish) = P(Ryan chooses Spanish) * P(Riley chooses Spanish)
= 0.86 * 0.73
= 0.628
Therefore, the probability that they both choose Spanish is 0.628.
b. To find the probability that neither Ryan nor Riley choose Spanish, we can use the complement rule. The complement of an event A is the probability that A does not occur:
P(neither choose Spanish) = 1 - P(at least one of them chooses Spanish)
Since we will calculate P(at least one of them chooses Spanish) in part d, we will save this calculation for later.
c. To find the probability that only one of them chooses Spanish, we need to consider two possibilities: either Ryan chooses Spanish and Riley does not, or Riley chooses Spanish and Ryan does not. Then we can add these two probabilities:
P(only one chooses Spanish) = P(Ryan chooses Spanish) * P(Riley does not choose Spanish)
+ P(Ryan does not choose Spanish) * P(Riley chooses Spanish)
Let's calculate this:
P(only one chooses Spanish) = (0.86 * 0.27) + (0.14 * 0.73)
= 0.232 + 0.102
= 0.334
Therefore, the probability that only one of them chooses Spanish is 0.334.
d. To find the probability that at least one of them chooses Spanish, we can use the complement rule again:
P(at least one chooses Spanish) = 1 - P(neither choose Spanish)
Now we can calculate it using the information we saved from part b:
P(at least one chooses Spanish) = 1 - P(neither choose Spanish)
= 1 - 0.334
= 0.666
Therefore, the probability that at least one of them chooses Spanish is 0.666.