Question 1180625: The probability P(x) of an event happening is equal to the number of ways N(x) an event can come out favorably, divided by the number of all possible ways S(x) the event can turn out, P(x)= N(x)/S(x). Managers of a baseball stadium decide to hand out progressively more free baseball caps to random attendees with each successive game, according to the function x+5, where x = 1 for the first game, x = 2 for the second game, and so on. This causes more fans to attend, according to the function 2x+3. Compose a probability function P(x) denoting the probability that a fan in attendance on day x will receive a baseball cap. State its domain and range, in practical terms (think about games and baseball caps).
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **Probability Function:**
The probability function P(x) that a fan in attendance on day x will receive a baseball cap is:
P(x) = (x + 5) / (2x + 3)
where:
* x = the game number (x = 1 for the first game, x = 2 for the second game, and so on)
* x + 5 = the number of baseball caps given away on day x
* 2x + 3 = the number of fans attending on day x
**Domain:**
In practical terms, the domain of the probability function is the set of positive integers representing the game number. Since the number of caps and attendees cannot be negative, and the game number must be a whole number, the domain is:
Domain: {x | x is an integer, x ≥ 1}
**Range:**
The range of the probability function represents the possible probabilities of a fan receiving a baseball cap. Since the number of caps given away is always less than the number of fans attending, the probability will always be between 0 and 1. As the game number increases, the probability approaches 1/2 since the number of caps and the number of attendees both grow linearly with x, but the number of attendees grows twice as fast.
Range: {P(x) | 0 < P(x) < 1}
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