Consider the letters of the word MERAFONG. How many unique word arrangements can be made using all the letters? Calculate the probability that the word arrangement will start with a letter R and end with a letter N.

There are 8 distinct letters in MERAFONG.

The number of unique arrangements is $8!$.

The number of unique word arrangements is $\boxed{40320}$.

11.1.2 Calculate the probability that the word arrangement will start with a letter R and end with a letter N.

Total arrangements = $8! = 40320$.

• The first letter must be R (1 choice).
• The last letter must be N (1 choice).
• The remaining 6 letters (M, E, A, F, O, G) can be arranged in the 6 middle positions in $6!$ ways. Number of favorable arrangements = $1 \times 6! \times 1 = 720$.

The probability is $\boxed{\frac{1}{56}}$.

11.1.3 Calculate the probability that the vowels (a, e, o) will be placed next to each other in any of the word arrangements.

Total arrangements = $8! = 40320$.

• Identify the vowels: A, E, O (3 vowels).
• Identify the consonants: M, R, F, N, G (5 consonants).
• Treat the 3 vowels as a single block. Now we are arranging 6 items: (AEO block), M, R, F, N, G. These 6 items can be arranged in $6!$ ways.
• Within the vowel block, the 3 vowels (A, E, O) can be arranged among themselves in $3!$ ways. Number of favorable arrangements = $6! \times 3! = 720 \times 6 = 4320$.

The probability is $\boxed{\frac{3}{28}}$.

11.2 The probability that during a given weekend a PSL team player will play in an international game (S) is 0.7. The probability that the player will play in a charity golf tournament (G) during that weekend is 0.4, and the probability that the player will participate in both activities is 0.3.
Given:
$P(S) = 0.7$
$P(G) = 0.4$
$P(S \cap G) = 0.3$

11.2.1 Draw a Venn diagram to represent the given information.

• Probability of S only: $P(S \text{ only}) = P(S) - P(S \cap G) = 0.7 - 0.3 = 0.4$.
• Probability of G only: $P(G \text{ only}) = P(G) - P(S \cap G) = 0.4 - 0.3 = 0.1$.
• Probability of S or G (union): $P(S \cup G) = P(S) + P(G) - P(S \cap G) = 0.7 + 0.4 - 0.3 = 0.8$.
• Probability of neither S nor G (outside both circles): $P(\text{neither}) = 1 - P(S \cup G) = 1 - 0.8 = 0.2$.

A Venn diagram consists of a rectangle representing the sample space (total probability 1). Inside, there are two overlapping circles, one for event S and one for event G.

• The overlapping region (intersection) contains the value 0.3.
• The part of circle S that does not overlap with G contains 0.4.
• The part of circle G that does not overlap with S contains 0.1.
• The region outside both circles but inside the rectangle contains 0.2.

11.2.2 Calculate:

a) P(S')

b) P(S and G)'

c) P(S or G)