(I wasn’t able to find this specific question, so sorry if it’s been asked before)
Earlier today, I was told I would get a call between 5:00 and 6:00. If we make the assumption that this call is, for certain, going to occur within the specified time frame, I was under the assumption that every minute that passed would result in an increase in probability of the call would happen. For example, I was thinking that for the first minute the chance I get the call are 1/60, while the second minute would be 1/59 (as there are now only 59 minutes in which it could happen) and so on until the last minute is 1/1 (if I had still not received the call).
However, I was thinking back to my Freshman statistics course where my professor was talking about how the chances of the event happening at any given moment are the same, regardless of what time it is. In this case, at any given minute, the chance of me getting a call would still be 1/60. Unfortunately, I don’t remember any of the terminology that goes with these concepts, so I can’t remember if that actually applies to this problem.
So, in short, does the probability of something presumably inevitable increase as time passes? Or is the chance the same at any given moment?
In: Mathematics
TLDR: You and your professor are both right. You’re playing different games.
Here’s a card game version of your problem: Take a standard 52-card deck and add 8 jokers (to get 60 cards). When the clock ticks to the next minute, you turn over 1 card. When you turn over the Ace of Spades, the call arrives.
Which of these three outcomes are possible?
– (a) You never draw the Ace of Spades.
– (b) You draw the Ace of Spades once.
– (c) You draw the Ace of Spades more than once.
“Which outcomes are possible?” is a trick question: I didn’t describe the rules clearly enough yet. I could actually be describing two different games:
– Your card game: After you draw a card, you discard it. You leave all the discards in a face-up pile, and don’t clean them up until the hour has passed. In total, you shuffle the deck once, at the beginning.
– Professor’s card game: After you draw a card, you shuffle it back into the face-down draw pile. You shuffle the deck 60 times, once for each draw.
In your card game, (a) (c) are impossible. You have to draw the Ace of Spades sometime. After the 60th draw, all the cards will be face-up, and the Ace of Spades must be among them. Once the Ace of Spades is face-up in the discard pile, it’s impossible for it to show up in future draws.
In the professor’s card game, you have a shuffled full deck every time. The Ace of Spades might or might not be the top card. (a) (b) (c) are all possible in the professor’s game.
Probability increases in your game, but not in the professor’s game.
More formally, your card game is “drawing without replacement,” and the professor’s card game is “drawing with replacement.”
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