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Probability

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If I choose four cards from a standard 52-card deck, with replacement, what is the probability that I will end up with four Aces?

Jan 16, 2022

#1
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If I choose four cards from a standard 52-card deck, with replacement, what is the probability that I will end up with four Aces?

Is this a trick question?  If every time you draw a card you put it back in the deck, you won't end up with any cards.

Jan 16, 2022
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Gracie Allen lives! Long Live Gracie...

See my solution below for pedantic and ultra-pedantic interpretations.

GA

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Jan 16, 2022
edited by GingerAle  Jan 16, 2022
edited by GingerAle  Jan 16, 2022
#3
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Here, I answer a very similar question. If I choose four cards from a standard 52-card deck, with replacement, what is the probability that I will end up with all four Aces?

This solution post is based on the original:

The phrase “with replacement” is an explicit standard variation in the sampling method. In this case, it means the card (no matter what it is) is replaced after it is drawn.  It’s also important to note that the deck or selection process remains randomized after the card is replaced. And, for the ultra pedantic minded (Ron), the value of the drawn card is recorded before its replacement into the deck. The records are then analyzed statistically for the probability of drawing four aces in sequence.

This question is poorly written. The primary defect is the phrase “end up with four Aces,” which is a colloquialism as used here. The capital “A” in “Aces” is nonstandard, and gives ambiguous emphases to the word aces. The word “all” is not used in this question, so the interpretation is biased toward any combination of Aces where one or more may be repeated.

Rephrased as primer statistics question:

If I choose four cards with replacement from a standard 52-card deck, what is the probability that I will select four aces (of any suit)?

A success in this experiment is four aces without regard to the suit.

$$\large \left(\dfrac{4}{52}\right)^4 = \dfrac{1}{28561}$$

Compare to the probability of selecting four aces in four different suits, where any other combination is a failure.

$$\large \dfrac{4}{52}*\dfrac{3}{52}*\dfrac{2}{52}*\dfrac{1}{52} =\dfrac{3}{913952}$$

GA

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GingerAle  Jan 16, 2022
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Good explanation.  I actually prefer "literal" but for you I'll cop to pedantic, negative connotation notwithstanding.

Guest Jan 16, 2022
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Good explanation.  I actually prefer "literal" but for you I'll cop to pedantic, negative connotation notwithstanding.

You don’t need to cop to pedantic; I do not think you are pedantic. You’re not pedantic in your solution presentations. (If you were, I’d be trolling the hell out of you.)  Although apparently, (sometimes) you tend toward the literal for certain colloquialisms (and Latin word-order in certain phrases), you do not seem to be a diehard literalist. I think this quite funny, and I’m sure Gracie Allen would find it funny and endearing.

If someone asked you to bring them a cup of coffee, what would you bring to them?

A cup of coffee beans, a cup of ground coffee beans, a cup of dehydrated coffee extract, or a cup of brewed or instant coffee?

When I was eleven, I once brought a literal cup of coffee (the mug was two-thirds full of instant coffee crystals) to my mum, after she asked me to bring her a cup of coffee. My mum was greatly amused. The idea came to me after my Great Uncle Cosmo explained certain food chemistries to me. One of the examples was that a typical cup of coffee was 1.13% to 1.88% soluble vegetable matter (extracted from approximately 12 grams of ground coffee beans) and the rest was water.  A cup of coffee is 98.75% water, but it’s called a cup of coffee. I never thought of it any other way until after my chemistry lesson from Cosmo.

GA

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GingerAle  Jan 17, 2022
edited by GingerAle  Jan 17, 2022
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Hello Ginger,  my mind is off on a very silly wander....

When people take 3 or more spoons of sugar in their coffee I often make a snide remark regarding them having coffee with their sugar.

You are right they are having sugar with their water. Maybe with a sprinkle of coffee for added flavour.

I suppose a person and many other things are largely composed of water.

Giving birth to water sounds somewhat less painful than giving birth to a baby.

Wandern, in German, means to hike.  Vielleicht könnte ich zu einem See wandern, und einen Fisch fangen!

Mmm Abendessen!

That sounds much more productive than this silly soliloquy is.  :)

Melody  Jan 17, 2022
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Hi Melody, I just noticed your post...

Vielleicht könnte ich zu einem See wandern, und einen Fisch fangen!

I’ll say that, in your wandering soliloquy, your comment is in perfectly composed German syntax.  I can read and speak German reasonably well at the social level. When the conversation becomes technical, such as in complex science or math, I’ll spend more time using a dictionary or translator than writing or speaking.

Once, while I was staying at a vacation rental near Bonn, the owners’ 22-year-old son, Hans, who was principal caretaker and bellhop, asked if I needed anything.  I wanted the bed moved a few feet closer to the window, so I could see the near full moon setting in the predawn morning. So, I said,

“Bitte hilf mir mein Bett in den Mondstrahlen zu erleben.”

What I should have said is,

"Bitte helfen Sie mir, mein Bett in den Mondstrahlen zu bewegen"

Hans could tell what I meant by my gestures and countenance, and I could tell he was greatly amused by my request.  He also said he’d be very happy to do both.

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When people take 3 or more spoons of sugar in their coffee I often make a snide remark regarding them having coffee with their sugar.

LOL!!  I make similar snarky remarks, and sometimes I ask if I can bring them insulin and a syringe. I usually reserve the insolent-insulin remarks for those who put a third-cup or more of sugar in a twelve ounce mug of coffee.

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Giving birth to water sounds somewhat less painful than giving birth to a baby.

Years ago, I read a science fiction story where the earth-visiting aliens referred to the humans as “big bags of mostly water.”   We are LOL!

GA

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GingerAle  Jan 23, 2022
edited by GingerAle  Jan 23, 2022