Question

There are 26 letters in the English alphabet. Letters are picked one by one at random, so that each letter has the same chance of appearing as any other letter, regardless of which letters have appeared or not appeared.
a. Six letters are picked. Find the chance that the sequence that appears is RANDOM, in that order.
b. Six letters are picked. Find the chance that they can be arranged to form the word RANDOM.

Answer 1

A coin lands either heads or tails. Unless the problem states otherwise, you can assume that the coin is fair, that is, the faces are equally likely.

A die is a cube; its six faces show 1, 2, 3, 4, 5, and 6 spots. Unless the problem states otherwise, you can assume that the die is fair, that is, all six faces are equally likely. The plural of “die” is “dice.”

A standard deck of cards consists of 52 cards. There are 13 cards in each of 4 suits: hearts, diamonds, spades, and clubs. Hearts and diamonds are red; spades and clubs are black. Each suit consists of the ranks Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen King.

“Cards are dealt from a well shuffled standard deck” means “cards are drawn at random without replacement.”

Cards are dealt from a well shuffled standard deck. Find the chance that:
a. the first card is not a heart
b. the 10th card is the ace of spades
c. the 10th card is the ace of spades, given that the 50th card is the ace of diamonds
d. the 32nd card is red, given that the 51st and 52nd cards are red
e. the 51st card is red, given that the 32nd and 52nd cards are red

Answer 2

The solutions to the first question are found as follows:
a. There is only one correct choice for each selection. The required probability is
\[P=\frac{1}{26}\times \frac{1}{25}\times \frac{1}{24}\times \frac{1}{23}\times \frac{1}{22}\times \frac{1}{21}\]
b. There are 6 choices of letter on the first selection, 5 choices on the next selection and so on. the required probability is
\[P=\frac{6}{26}\times \frac{5}{25}\times \frac{4}{24}\times \frac{3}{23}\times \frac{2}{22}\times \frac{1}{21}\]

Answer 3

@kropot i think it is with replacement, not without , but could be reading it wrong

Answer 4

Cards are dealt from a well shuffled standard deck. Find the chance that:
a. the first card is not a heart
one quarter of the cards are hearts and three quarters are not
your answer is \(\frac{3}{4}\)

Answer 5

b. the 10th card is the ace of spades
same as the probability the first card is the ace of spades, there is only one out of the total of 52, answer is \(\frac{1}{52}\)

Answer 6

c. the 10th card is the ace of spades, given that the 50th card is the ace of diamonds

now you only have 51 cards to choose from, since one is already at the bottom of the deck, so your answer is \(\frac{1}{51}\)

Answer 7

d. 24/50
e. 24/50

Answer 8

many thnks

Answer 9

A person is selected at random from a population that has the following characteristics:
65% of the people are men; the others are women
12% of the men are smokers
7% of the women are smokers

a. Find the chance that the selected person is a non-smoker, given that the person is a man.
b. Find the chance that the selected person is a woman who smokes.
c. Find the chance that the selected person is a smoker.
d. Given that the selected person is a smoker, what is the chance that the person is a woman?
e. Find the chance that the selected person is a man or a non-smoker.

Answer 10

@satellite73 Regarding your comment on the first question, I think the second part must be sampling without replacement the reason being that the six letters that are picked can be arranged if appropriate. By inference the first part of the question could be also intended to be sampling without replacement, but the question is not clear on this point.

Answer 11

So, there is any answer of this Q?

Answer 12

There are 26 letters in the English alphabet. Letters are picked one by one at random, so that each letter has the same chance of appearing as any other letter, regardless of which letters have appeared or not appeared.

PROBLEM 5A

This Problem is worth 1 Point
Six letters are picked. Find the chance that the sequence that appears is RANDOM, in that order.

Answer 13

After reconsideration of the RANDOM question and the post by @satellite73, I am now convinced that part a is sampling with replacement.
Therefore the chance that the sequence that appears is RANDOM, in that order is given by
\[1\ out\ of\ 26^{6}\]

Answer 14

kropot72, I think it has to be (1/26)(1/25)(1/24)(1/23)(1/22)(1/21) rather than (1/26)^6

Answer 15

@chrisindallas d answer is (1/26)^6

Answer 16

its without replacement, because you just have 26 letters to choose from