6. what is the total number of different 9 - letter arrangements that can be formed using the letters in the…

6. what is the total number of different 9 - letter arrangements that can be formed using the letters in the word tennessee?\n7. 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?\n8. 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)=\n9. 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))\n10. true or false? if p(a and b)=p(a)×p(b), then we can classify two events as independent events. circle one.

6. what is the total number of different 9 - letter arrangements that can be formed using the letters in the word tennessee?\n7. 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?\n8. 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)=\n9. 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))\n10. true or false? if p(a and b)=p(a)×p(b), then we can classify two events as independent events. circle one.

Answer

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