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Two psychologists are studying the IQs of a group of business professionals.

- Let [tex]$A$[/tex] be the event that a professional's IQ is greater than 130, with [tex]$P(A)=16\%$[/tex].
- Let [tex][tex]$B$[/tex][/tex] be the event that a professional's IQ is between 115 and 130, with [tex]$P(B)=14\%$[/tex].

The IQs of different professionals are independent events. What is the probability that two random professionals will be chosen and one of their IQs is greater than 130 and the other is between 115 and 130?

Give your answer as a decimal, rounded to three decimal places.

Sagot :

GinnyAnswer
Sure, let's solve this problem step-by-step.

### Step 1: Identify the Probabilities
First, we know the following probabilities:
- [tex]\(P(A)\)[/tex]: Probability that a business professional's IQ is greater than 130 is 0.16.
- [tex]\(P(B)\)[/tex]: Probability that a business professional's IQ is between 115 and 130 is 0.14.

### Step 2: Calculate the Probability for One Specific Order
Since the IQs of different professionals are independent events, we can multiply these probabilities to find the probability that the first professional's IQ is greater than 130 and the second professional's IQ is between 115 and 130, i.e., [tex]\(P(A) \times P(B)\)[/tex]:
[tex]\[ P(A \text{ and } B) = 0.16 \times 0.14 = 0.0224 \][/tex]

### Step 3: Consider Both Orders
The above calculation only handles one specific order. We must also consider the reverse order: the first professional's IQ is between 115 and 130 and the second's is greater than 130. This is also an independent event and has the same probability:
[tex]\[ P(B \text{ and } A) = 0.14 \times 0.16 = 0.0224 \][/tex]

### Step 4: Add the Probabilities Together
Since either order is acceptable, we add these probabilities to get the total probability that one professional has an IQ greater than 130 and the other has an IQ between 115 and 130:
[tex]\[ P(\text{one's IQ} > 130 \text{ and the other's IQ} \text{ between 115 and 130}) = 0.0224 + 0.0224 = 0.0448 \][/tex]

### Step 5: Round the Answer to Three Decimal Places
Finally, we round the result to three decimal places:
[tex]\[ \boxed{0.045} \][/tex]

Therefore, the probability that one professional has an IQ greater than 130 and the other has an IQ between 115 and 130 is 0.045.
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