Question

1. what is the total number of different 9 - letter arrangements that can be formed using the letters in the word tennessee?
2. two cards are drawn together from a pack of 52 cards without replacement. what is the probability of drawing one heart and one spade regardless of the order that you draw them in?
3. a computer program will randomly select a digit from the set of numbers {1, 3, 4, 6, 7, 9}. what is the probability that the digit selected at random will not be a multiple of 3? p (not a multiple of 3)=
4. consider events a and b, where the events are independent of each other. if p(a)=0.35 and p(b)=0.6, what is probability of a and b occurring together? (p(a and b))
5. true or false? if p(a and b)=p(a)×p(b), then we can classify two events as independent events. circle one.

Explanation:

Step1: Calculate arrangements of TENNESSEE

The word TENNESSEE has 9 letters with 1 T, 4 E's, 2 N's and 2 S's. The formula for permutations of a multi - set with \(n\) objects where there are \(n_1,n_2,\cdots,n_k\) indistinguishable objects of different types is \(\frac{n!}{n_1!n_2!\cdots n_k!}\). Here \(n = 9\), \(n_1=1\) (for T), \(n_2 = 4\) (for E), \(n_3=2\) (for N) and \(n_4 = 2\) (for S). So the number of arrangements is \(\frac{9!}{1!4!2!2!}=\frac{9\times8\times7\times6\times5\times4!}{1\times4!\times2\times1\times2\times1}=3780\).

Step2: Calculate probability of drawing one heart and one spade

There are 13 hearts and 13 spades in a 52 - card deck. The number of ways to choose one heart and one spade (without order) is \(C(13,1)\times C(13,1)=13\times13 = 169\). The number of ways to choose 2 cards from 52 without replacement is \(C(52,2)=\frac{52!}{2!(52 - 2)!}=\frac{52\times51}{2\times1}=1326\). The probability is \(\frac{169}{1326}=\frac{13}{102}\). The given \(\frac{13}{52}\times\frac{13}{51}\) is incorrect as it doesn't account for the two possible orders (heart then spade or spade then heart).

Step3: Calculate probability of non - multiple of 3

The set is \(\{1,3,4,6,7,9\}\). There are 6 numbers in the set. The multiples of 3 in the set are 3, 6, 9. So there are 3 non - multiples of 3. The probability \(P(\text{not a multiple of }3)=\frac{3}{6}=\frac{1}{2}\), which is correct.

Step4: Calculate \(P(A\cap B)\) for independent events

For independent events \(A\) and \(B\), the formula is \(P(A\cap B)=P(A)\times P(B)\). Given \(P(A)=0.35\) and \(P(B)=0.6\), then \(P(A\cap B)=0.35\times0.6 = 0.21\).

Step5: Determine truth of independent events definition

If \(P(A\cap B)=P(A)\times P(B)\), then by the definition of independent events, \(A\) and \(B\) are independent events. So the statement is True.

Answer:

1. 3780
2. \(\frac{13}{102}\)
3. \(\frac{1}{2}\)
4. 0.21
5. True