It's Only a Flesh Wound

Probability Level 4

King Arthur is in combat with The Black Knight. The Black Knight has four limbs; two arms and two legs. Arthur chooses a limb to strike at randomly and has a 50% chance of cutting the limb off.

The Black Knight will only be stopped when all of his limbs have been removed.

What is Arthur's probability of victory in 10 strikes? (Round your answer to the tenths place.)

If you're unfamiliar with this classic scene, it can be seen here .

Details and Assumptions

  • The loss of more limbs of The Black Knight does not increase King Arthur's chance of cutting the remaining limbs off.


The answer is 0.8.

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Brock Brown by Brock Brown , 25, USA

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3 solutions

Philip Lamkin
Feb 15, 2015

Arthur can lose if he only hits 1, 2 or 3 times out of the 10 strikes. He can hit once in ( 10 1 ) = 10 {10 \choose 1} = 10 ways, twice in ( 10 2 ) = 45 {10 \choose 2} = 45 ways, or three times in ( 10 3 ) = 120 {10 \choose 3} = 120 ways. In total, this is 175 ways Arthur can lose out of the 1024 total possibilities. 1024 − 175 1024 ≈ 0.829 \frac{1024-175}{1024} \approx 0.829 , which gives us our answer to the nearest tenth as 0.8 \boxed{0.8}

6 Helpful 0 Interesting 0 Brilliant 0 Confused

he could also get hit 0 times (1 way). so it's .828125

William Torres - 6 years, 3 months ago
Brock Brown
Feb 12, 2015

A Monte Carlo simulation for the classic Monty Python scene: 

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from random import random
def victory():
    limbs = 4
    strikes = 10
    for strike in xrange(strikes):
        if random() < 0.5:
            limbs -= 1
        if not limbs:
            return True
    return False
yes = 0.0
fights = 1000000
for fight in xrange(fights):
    if victory():
        yes += 1
print "Answer:", round(yes/fights,1)

3 Helpful 0 Interesting 0 Brilliant 0 Confused

FYI - You can edit your problem through the "dot dot dot" menu and selecting "edit problem". There is currently no way for you or the moderators to edit the image of your problem (the one that is displayed in the main condensed view).

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Calvin Lin Staff - 6 years, 4 months ago

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Okey doke, thanks Calvin.

Brock Brown - 6 years, 4 months ago
Linchuan Zhang
Feb 15, 2015

Assumptions:

1)King Arthur will only choose to strike limbs that have not already been struck. 2)King Arthur will not be killed or otherwise incapacitated by the Black Knight before he makes his 10 strikes.

Solution: Since the probability of striking off each limb is identical, we can basically treat every limb as equal.

Thus, the equation reduces (to an admittedly macabre) coin flip question

What is the probability of getting at least 4 heads if you flip 10 fair coins?

By the Law of Total Probability (http://en.wikipedia.org/wiki/Law of total_probability), this is equivalent to

1-[P (King Arthur is an utter failure and chops off nothing) +P(King Arthur chopping off only one limb) +P(King Arthur chopping off exactly two limbs) +P(King Arthur chopping off exactly three limb)]

Now that the problem is so presented, the solution is trivial:

P(0)=1/1024 P(1)=10/1024 P(2)=45/1024 P(3)=120/1024

P(0Λ1Λ2Λ3)=176/1024 1-176/1024=0.828125~=.8

QED

PS. This question is not very well formulated, IMHO. It can easily be altered to address my two assumptions.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

I felt that those two assumptions were fair. This is because when the limb is chopped off there no longer exists a limb to strike at. Maybe the assumption that King Arthur had no chance of being struck was a little unfair, but I wanted to be accurate to the original film. He was never struck by The Black Knight's blade, although he did receive a few good solid kicks.

Also, solid solution. I didn't mean to critisize the value of your solution by standing behind my assumptions, good job. Upvoted.

For fun though let's assume that King Arthur has a 10% chance of death each round because he's a skilled swordsman, but he lacks the ability to stay alive after being struck. This would reduce his odds of victory to about 40%. (Note that this assumes that the Black Knight strikes first.)

If you make the even further assumption that King Arthur might strike at a limb that he already removed, then the odds are reduced to 10%. Here is the modified simulation: 

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from random import random, choice
def victory():
    limbs = 4
    strikes = 10
    struck = ''
    for strike in xrange(strikes):
        if random() < 0.1:
            return False
        if random() < 0.5:
            aim = choice('1234')
            if aim not in struck:
                struck = struck + aim
                limbs -= 1
        if not limbs:
            return True
    return False
yes = 0.0
fights = 100000
for fight in xrange(fights):
    if victory():
        yes += 1
print "Answer:", yes/fights

Edit: I've edited this comment A LOT since I first posted it because I had to make a lot of corrections, but I think it's right now.

Brock Brown - 6 years, 3 months ago

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Sorry, I didn't see this comment until just now. Clever computer simulation. ;-)

Yeah, assumption 1 was really pedantic on my part.

Linchuan Zhang - 6 years, 1 month ago

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