6. [Counting, 10 points] Suppose that are considered), with repeated letters all

Question

6. [Counting, 10 points] Suppose that are considered), with repeated letters allowed. a "word" is any string of seven letters of the alphabet (only uppercases (2) How many words beg hnd nith 17 (3 points) (2) How many words begin with R and end with T? [3 points (3) How many words begin with a vowel or end with a vowel? [4 points) 7. [Probability, 10 points] (1) You flipped a coin 200 times. Out of these 200 outcomes, there were 80 of them head appeared. What is the probability to get a tail, p(tail), when you flip this coin? [4 points] (2) Suppose that you have another coin, which tail is four times as likely to appear as head. Find the probability of head and tail, p(head) and p(tail). [4 points] (3) Considered the coins in (1) and (2), which one is less biased? [2 points

Explanation / Answer

6) as there are 26 choices are there for each letter;

1) number of ways =267  =8031810176

2)for frst and last word are fixed ; we need to choose in between 5 letters ; for whcih number of ways =265

=11881376

3) for first letter to be vowel number of ways =5*266 =1544578880

for last letter to be vowel number of ways =5*266 =1544578880

for first and last letter to be vowel number of ways =5*5*265 =297034400

hence totoal ways in which first or last letter is vowel =1544578880+1544578880-297034400 =2792123360

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