Statistical Physics

1) The Lab Manual shows that if you randomly drop sticks of length L on a
floor covered with square tiles of width W, the average value of the
*probability* that a stick touches or crosses the edge of a tile is

<T>= 4 L/(p W). | ...... | (1) |

This can be rearranged as

p = 4 L/(<T> W). | ...... | (2) |

A student measured the values L=16.0 ±
0.01 cm and W= 25.0 ±
0.05 cm. Then the student* experimentally measured* the fraction of
dropped sticks which touched or crossed a line, and substituted this for the
*probability* <T>. The resulting value of the ratio in (2) was
2.95, which does not agree very well with the value of
p= 3.14159…... . *Which of the
following do you think is the most likely source of the disagreement?*

- L was not measured with sufficient accuracy.
- W was not measured with sufficient accuracy.
- Not enough sticks were dropped and counted.
- The student must have made an arithmetic error.
- None of the above.

*Justify* your choice.

**2) **Four "honest" coins are flipped once each.
The *probability* of getting 1 head and 3 tails (1H and 3T) equals
the number of ways to achieve this combination, divided by the number of
possible outcomes.

Pick the set of completions which make the following statement completely correct.

There are ______ possible outcomes for flipping 4 coins, there are ______ ways to achieve the combination of 1 H and 3 T, and the probability of getting 1 H and 3 T is ________.

- 4; 4; 4/4
- 8; 4; 4/8
- 8; 8; 8/8
- 16; 4; 4/16
- 16; 5; 5/16

Option E is not correct. Explain exactly what is wrong with it. (Don't
merely identify the incorrect number or numbers; explain how the
*correct* numbers are obtained.)

**3)** **Pick the set of completions which make the following statement
completely correct.**

Five "honest" coins are flipped once each. The probability of getting 1 head and 4 tails (1H and 4T) is ______, while the probability of getting 2 heads and 3 tails (2H and 3T) is _________.

- 1/5; 2/5
- 5/16; 5/16
- 5/16; 10/16
- 5/32; 5/32
- 5/32; 10/32

Pick the correct option, and show how you obtain your result.

**4) **Consider two coins that we assume to be "honest". The coins are to be flipped** **100,000 times.
** **The *probability* for getting 1 head and 1 tail (1H and 1 T)
______________.

- depends on the number of flips
- cannot be determined before performing this experiment
- would be smaller if only 1,000 flips were performed
- depends on the assumption that the coins are "honest"
- is 0.25

*Justify* your choice.

**5) **Two "honest"** **coins were flipped 20 times.
The graph shows F, the fraction of two-coin flips, versus the number of heads
H that resulted. *Pick the false statement.*

A) The individual fractions must add up to a total of 1.00 or 100%.

B) The experimental distribution does not agree with the probability
distribution.

C) If the experiment is continued to a very large number of flips, we would
expect the experimental distribution to approach agreement with the
probability distribution.

D) For only 20 two-coin flips, it would not have been surprising if the
fraction of flips giving 1 Head and 1 Tail were less than 0.5.

E) The experimental result must be wrong; the fraction with H = 0 must equal
the fraction with H = 2.

*
*

*Justify* your choice.

**6)** Two "honest"** **coins were flipped 20 times. The
graph shows F, the fraction of two-coin flips, versus the number of heads H
that resulted. *Pick the false statement.*

A) The experimental result must be wrong; the individual fractions must
add up to a total of 1.00 or 100%.

B) The experimental distribution does not agree with the probability
distribution.

C) If the experiment is continued to a very large number of flips, we would
expect the experimental distribution to approach agreement with the
probability distribution.

D) For only 20 two-coin flips, it is not surprising if the fraction of flips
giving 1 Head and 1 Tail is less than 0.5.

E) The experimental fraction with H = 0 must equal the fraction with H = 2.

*Justify* your choice.