Old questions are often fun.

[theweaselking]

A trio of very old but still very neat questions.

1. A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?

2. If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?

3. In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

Responses screened, initially!

ETA: Ooh, a 4th.

4. A certain town is served by two hospitals. In the larger hospital about 45 babies are born each day, and in the smaller hospital about 15 babies are born each day. As you know, about 50% of all babies are boys. However, the exact percentage varies from day to day. Sometimes it may be higher than 50%, sometimes lower.
For a period of 1 year, each hospital recorded the days on which more than 60% of the babies born were boys. Which hospital do you think recorded more such days?

a)The larger hospital
b)The smaller hospital
c) About the same (that is, within 5% of each other)


EDIT: OKAY IT'S BEEN 14 HOURS. ANSWERS ARE UP. Feel free to guess if you haven't read the answers yet.

Comments

[a-pawson.livejournal.com]

1. $0.05
2. 5 minutes
3. 47 days
4. The smaller hospital

[scifantasy.livejournal.com]

1) 5 cents; the ball costs 5 cents, the bat a buck five, making the total a buck ten.
2) 5 minutes. Each machine makes one widget in five minutes.
3) 47 days. If it doubles in size, then the day before it covers the entire lake, it covers half.
4) c, about the same. This one, I just think is right. I mean, it's as likely that the percentage is over 60 at each place, yeah?

[davidcook.livejournal.com]

1. $0.05
2. 5 minutes
3. 47 days
4. c.

[bart-calendar.livejournal.com]

The first one is nine cents.

[argonel.livejournal.com]

1. The ball costs $0.05
2. It takes 100 machines 5 minutes to produce 100 widgets
3. It takes 47 days for the lilly pads to cover half of the lake.
4. The smaller hospital records more days where more than 60% of births are boys, but I'm not sure if it would be by more than 5%

[calankh.livejournal.com]

1. $0.05
2. 5 minutes
3. 47 days
4. b

i'm too lazy to do the binomial calculation for 4 so that's just a guess.

[tisiphone.livejournal.com]

1. The ball costs five cents.
2. Five minutes.
3. 47 days.
4. The answer you're looking for is c. However, this could actually be more complex depending on the distribution of births, the sample size, etc. In general, smaller sample sizes show more volatility.

[theweaselking.livejournal.com]

I figured I'll unscreen answers here, and post the correct answers in a new post with a link back, sometime this evening. Give people 10-12 hours to play.

PS: Anyone who wants to discuss or argue before then, just ask and I'll unscreen your comment and any responses.

[jsburbidge.livejournal.com]

a) 5 cents
b) 5 minutes
c) 47 days
d) c)

[ironphoenix.livejournal.com]

1. $0.05
0.05 + ($1.00 + $0.05) = $1.10

2. 5 min.
5 widgets/5 machines/5 min = 1 widget/1 machine/5 min
= 100 widgets / 100 machines/5 min.

3. 47
if K*2^48 = 1, then (K*2^48)/2 = 1/2, so K*2^(48-1) = 1/2, or K*2^47 = 1/2.

4. b) the smaller hospital
The larger hospital will have more regression to the mean.

Feel free to unscreen this!

[kjn]

1. $0.05.

2. Assuming each machine is working on one widget until it's finished, it will take 5 minutes.

3. 47 days.

4. c.

[kytheraen.livejournal.com]

1. 5 cents
2. 5 minutes
3. 47
4. C? Same chance, I guess.

[ariaflame.livejournal.com]

1. $0.05
2. 5 minutes.
3. 47 days

4. Not entirely sure. Still working on my statistics, inclined slightly towards smaller hospital because the same absolute variations there will make larger relative variations. Definitely not a)

[martinl-00.livejournal.com]

5cents, 5mins, 47days, and I can't be arsed to bust out the binomial theorem to check b) vs. c) ...

... dammit. OK, I gotta know now...

Cheating by using a web based calculator, it seems 9+ out of 15 flips is .30 or ~110 days and 27+ out of 45 flips is .12 or ~44, so b).

But it's gotta be a trick.

(To teh Googles)

Oh, it's just checking to see if the respondent is familiar with the effects of increasing sample size. I was looking for a trickier trick.

[ambug666.livejournal.com]

1,2, and 3 I've encountered before.

1. bat is 60 cents, ball is 50 cents. Every fiber in my being wanted to say the ball cost 10 cents even though I knew it was wrong.
2. 5 minutes
3. 47 days.

4 I'll have to cogitate on.

[drcuriosity]

1. The ball costs 5c, the bat costs $1.05.
2. 5 minutes. 20x manufacturing capacity can make 20x the output in the same amount of time.
3. 47 days.
4. c) - size of the hospital shouldn't affect the proportional variance, only the absolute number of babies required to reach the 60% threshold.

[anton-p-nym.livejournal.com]

Okay, I'll bite.

1. $1.05

2. 5 minutes

3. 47 days.

4. b), as a pure guess.

-- Steve looks forward to seeing the results.

[mr-rubix.livejournal.com]

I'll go ahead and try:

1) $0.05
2) 5 minutes
3) 47 days
4) B

[marcus-felix.livejournal.com]

1. $0.05
2. 5 minutes
3. 47 days
4. b

[skington.livejournal.com]

1: the ball costs $0.05, the bat costs $1.05, total $1.10
2: 5 minutes; that's how long it takes any one machine to make a widget, and you're running 100 of them at once.
3: 47 days
4: gut feeling is b, because it only takes 1 or 2 babies to get to 60% in the small hospital. (The small hospital also records more days where there are 40% male or fewer.) I'm now going to brute force it to see if I'm right.

ETA: yup, it's b.

[being-here.livejournal.com]

I think I am just going to be exposing my ignorance here, but:

1) $0.05
2) Five minutes
3) 47 days
4) The smaller hospital (because statistical variations are more noticeable in a smaller sample group)

[fuster-cluck.livejournal.com]

1. (ten cents)
2. (five minutes)
3. (47 days)
4. (the same)

[malovich.livejournal.com]

1) $0.05
2) 5 minutes
3) 47 days
4) b, the smaller hospital: a single bit of variance has far more impact (and thus more likely to go over the 60% threshold) than the larger pool of babies born in the larger hospital.

[jacob sewell (from livejournal.com)]

1: $.05
2: 5 minutes.
3: 47 days
4: The smaller hospital (b).

[andrewducker]

1) $0.05
2) 5 minutes
3) 47 days
4) (b) - the smaller hospital. Hitting 60% requires less individual events, and is thus statistically more likely.

[thornae.livejournal.com]

Okay, so the first three required but seconds of thought (probably in part because I've seen very similar versions of them before), but the fourth was new, and I enjoyed thinking about it. Yay maths!


So, in short order:
1. $0.05 (Edited to correct brainfart b/c I read "bat" instead of "ball").
2. 5 minutes
3. 47 days

and, although I'm less sure about this one because I can't prove it in the strictest sense,

4. The smaller hospital.

Reasoning for 4: Lower numbers mean greater statistical variance. To consider an extreme example, substitute in a hospital with about 2 births/day vs one with ~1000, and ask which one has more days of 100% boy births recorded.

Looking forward to seeing how many people get caught out...



See also Richard Wiseman's regular Friday Puzzle (http://richardwiseman.wordpress.com/2014/06/23/answer-to-the-friday-puzzle-263/#more-8664), and this hoary old favourite (http://www.shoptcs.com/cjp/rid/7.htm).

[endotoxin.livejournal.com]

Oh yeah, I'm game.

1: The bat costs $1.05, the ball $0.05.
2. 5 minutes.
3. 47 days.
4. C.

[snobahr.livejournal.com]

1. $0.10
2. 5 minutes.
3. 47 days.
4. C, because we're talking about percentages of a whole, not actual numbers.

[brithistorian.livejournal.com]

1. 5 cents.

2. 5 minutes.

3. 47 days.

4. B

[captainweasel.livejournal.com]

1. ball costs 0.05
2. 5 minutes
3. 47 days
4. c

[kierthos.livejournal.com]

1) $0.05 cents.

2) 5 minutes.

3) 47 days.

4) The smaller hospital.

[krfsm.livejournal.com]

1. $0.05. (($1.10 - $1)/2)
2. 5 minutes. (Same number of widgets/machine.)
3. 47 days. ("Every day, the patch doubles in size.")
4. c, about the same. (Given no other info, the distribution should be the same for both, but I suspect this is some kind of "gotcha" question.)

[lurkerwithout.livejournal.com]

#1 is five cents

#3 is 47 days

[frith]

1. 10 cents
2. 5 minutes
3. 47 days
4. c)

[jack-ryder.livejournal.com]

Okay, I'll bite

1) the ball costs 5c
2) 5 minutes
3) 47 days
4) I sense this is a trap but I'm going with b)

[prk.livejournal.com]

1) 5c (obvious but incorrect answer = $1)

2) 5 minutes (obvious but incorrect answer = 100 minutes)

3) 47 days (obvious but incorrect answer = 24 days)

4) Gut feeling is b) the smaller hospital, based on a lower sample size allowing a bigger swing from day to day, but I don't have time to do the maths, and my gut feeling may be the obvious but incorrect answer in this case.

prk.

[dianavilliers.livejournal.com]

1) $ 0.05
2) 5 minutes
3) 47 days (it must have been a very small patch to begin with, for any reasonable definition of lake)
4) The smaller.

[falconwarrior.livejournal.com]

1. 5 cents. .05 + 1.00 = 1.05, and 1.05 + .05 = 1.10.
2. 5 minutes. it seems to take 1 machine 5 minutes to make 1 widget so 100 machines could each make 1 widget in 5 minutes for 100 widgets.
3. 47 days. it will then take one day to get to day 48 when the lily pads will have doubled to cover the whole lake.
4. i don't really have a way to solve this one with confidence but my best guess is c.

[nurseysarah.livejournal.com]

1) 10 cents
2) 5 minutes
3) 47 days
4) no bloody idea

[skington.livejournal.com]

Or a very scary lily pad.

[theweaselking.livejournal.com]

I think any pad that doubles in size daily, regardless of previous size, is a VERY scary pad.

[frith]

Pouah. 50%. But it was still fun. 8^D

[dianavilliers.livejournal.com]

But think of the carbon sequestration possibilities.