Probability of a-b > 3 in Randomly Selected Intervals

In summary, the problem involves finding the probability that the difference between two randomly selected points, a and b, on the real line within specific intervals will be greater than 3. The probability can be determined by finding the fraction of all possible values within the given intervals that satisfy the condition a-b>=3, which can be visualized as a rectangle in a two dimensional plane. The line that bounds this region can be used to calculate the probability.
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Homework Statement



On the real line (-inf, inf), point a resides in the interval 0<=a<=4, and point b is in the interval -3<=b<=0; otherwise, points a and b are selected at random within their intervals. Find the probability that a-b is greater than 3.

Homework Equations





The Attempt at a Solution



So, 0<=a<=4 and -3<=b<=0 then 0<=a-b<=7. So is the probability that a-b is greater than 3, 4/8? There are 8 chances and only 4 probabilities that it could happen, 4,5,6,7.
 
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  • #2
Your choices aren't limited to integers. Draw the region of possible values of a and b in a two dimensional plane where one axis is a and the other is b. It's a rectangle, right? What fraction of that rectangle will satisfy a-b>=3? Find the line that bounds that region.
 
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FAQ: Probability of a-b > 3 in Randomly Selected Intervals

What is the meaning of "a-b > 3" in terms of probability?

The expression "a-b > 3" refers to the probability that the difference between two events, represented by a and b, is greater than 3. This means that there is a higher chance of the difference being larger than 3 compared to it being 3 or smaller.

How is the probability of "a-b > 3" calculated?

The probability of "a-b > 3" can be calculated using the formula P(a-b > 3) = P(a) + P(b) - P(a and b), where P(a) and P(b) represent the individual probabilities of events a and b, and P(a and b) represents the probability of both events occurring together.

What factors influence the probability of "a-b > 3"?

The probability of "a-b > 3" can be influenced by various factors such as the individual probabilities of events a and b, the relationship between the two events, and any external factors that may affect the occurrence of these events.

Can the probability of "a-b > 3" ever be negative?

No, the probability of "a-b > 3" cannot be negative. Probability is always a positive value between 0 and 1, where 0 represents impossibility and 1 represents certainty. Therefore, the probability of "a-b > 3" will always be equal to or greater than 0.

How can "a-b > 3" be used in practical applications?

The probability of "a-b > 3" can be used in various practical applications, such as in statistical analysis, risk assessment, and decision making. It can also be used to determine the likelihood of certain events occurring and to make predictions about future outcomes.

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