Explanation of task from the probability quiz

Hello, I studied this quiz.

Which of these events is the most likely to happen when flipping a fair coin?

Flip 2 or more heads when flipping 3 coins.

Flip 20 or more heads when flipping 30 coins.

Flip 200 or more heads when flipping 300 coins.

Instead of making an explicit calculation, think about how likely or unlikely each of these outcomes would be.

Explanation Correct answer: 2 out of 3

When you flip some number of coins, you expect about half of them to be heads. As the number of coins increases, you should expect the observed proportion of heads to be closer and closer to 50\%.50%. The casual logic is: sure, weird things can happen a few times, but not hundreds of times.

With more probability experience, you can formalize and quantify the ideas behind this problem.

Here are the actual probabilities for completeness:

P(\text{2 or more heads out of 3})=0.5P(2 or more heads out of 3)=0.5

P(\text{20 or more heads out of 30})\approx 0.05P(20 or more heads out of 30)≈0.05

P(\text{200 or more heads out of 300})\approx 0.000000004P(200 or more heads out of 300)≈0.000000004

And i can't find the explanation , how were those probabilities calculated? In this explanation i can only see formula and result

Note by Paul Gavrilov
11 months, 2 weeks ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

See the binomial distribution.

Example for 33 flips:

If NN is the number of heads after 3\color{#3D99F6}3 flips, then NB(3,0.5)N\sim B(\color{#3D99F6}3\color{#333333}, 0.5). Using the cumulative distribution (on your calculator, for 30 or 300 flips), Pr(N2)=Pr(N=2)+Pr(N=3)=(32)(0.5)2(10.5)32+(33)(0.5)3(10.5)13=38+18=12\text{Pr}(N\geqslant 2)=\text{Pr}(N=2)+\text{Pr}(N=3)=\binom{3}{2}(0.5)^{2}(1-0.5)^{3-2}+\binom{3}{3}(0.5)^3(1-0.5)^{1-3}=\dfrac{3}{8}+\dfrac{1}{8}=\color{#20A900}\boxed{\dfrac{1}{2}}.

Alternatively, Pr(n or more heads from k flips)=12kr=nk(kr)\text{Pr(n or more heads from k flips)}=\frac{1}{2^k}\sum_{r=n}^{k}\binom{k}{r}

As there are 2k2^k possible and equally likely outcomes (2 for the first flip, 2 for the second flip, ..., 2 for the kkth flip) and (kr)\binom{k}{r} ways to get rr coins.

Matthew Christopher - 11 months, 2 weeks ago

Log in to reply

I know.

I get this question wrong as well!

Yajat Shamji - 11 months, 2 weeks ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...