Question

A biased coin (one that is not evenly balanced) is tossed 6 times. The probability of Heads on any toss is

0.3. Let X denote the number of Heads that come up.

1. Does this experiment meet the requirements to be considered a Bernoulli Trial? Explain why or why

not.

2. If we call Heads a success, what would be the parameters of the binomial distribution of X?

(Translation: find the values of n and p)

3. What is the shape of this binomial distribution (right skewed, symmetric, left skewed)? Explain how

you know this.

4. What would be the random-variable notation to represent that exactly 2 Heads are tossed?

5. Using the Binomial Probability Formula and your answers from

questions 1-4, calculate the following:

P(X = 2)

P(1 < X < 5)

*NOTE: Round final answers to 3 decimal places.*

6. Find the mean and standard deviation of X.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

A new drug trial is being tested. The results of the trial show that 90% of the participants are cured of

their disease after using the new drug. (Assume that this does meet the requirements of Bernoulli

Trials.) Suppose you are studying the results of 3 people who participated in the trial.

1. Draw a tree diagram to show the possible outcomes of this study of 3 participants. Be sure to include

the probabilities for each success or failure on the branches of your tree. Include a listing of the

outcome and the probability of each outcome.

2. Construct a probability distribution table to show the number of participants cured of their disease

Answer #1

1. Yes as all the tosses are independent of each other. Also, each toss has just two outcomes, which satisfies the Bernoulli condition.

2. Here n=6 and p=0.3

3. This a right-skewed data.

4. If two heads are tossed then X=2.

5.

6. E(X) = np = 6*0.3 = 0.18

1. Here is the decision tree:

2.

X=x | 0 | 1 | 2 | 3 |

P(X=x) | 0.001 | 0.027 | 0.243 | 0.729 |

A balanced coin is tossed 3 times, and among the 3 coin tosses,
X heads show. Then the same balanced coin is tossed X additional
times, and among these X coin tosses, Y heads show.
a. Find the distribution for Y .
b. Find the expected value of Y .
c. Find the variance of Y .
d. Find the standard deviation of Y

A balanced coin is tossed 3 times, and among the 3 coin tosses,
X heads show. Then the same balanced coin is tossed X additional
times, and among these X coin tosses, Y heads show.
a. Find the distribution for Y .
b. Find the expected value of Y .
c. Find the variance of Y .
d. Find the standard deviation of Y

An ordinary (fair) coin is tossed 3 times. Outcomes are thus
triples of "heads" (h) and "tails" (t) which we write hth, ttt,
etc. For each outcome, let R be the random variable counting the
number of heads in each outcome. For example, if the outcome is
ttt, then =Rttt0. Suppose that the random variable X is defined in
terms of R as follows: =X−R4. The values of X are thus:
Outcome
tth
hth
htt
tht
thh
ttt
hht
hhh...

Find the probability of more than 30 heads in 50 flips of a fair
coin by using the normal approximation to the binomial
distribution.
a) Check the possibility to meet the requirements to use normal
approximation (show your calculation)
b) Find the normal parameters of the mean(Mu) and standard
deviation from the binomial distribution.
c) Apply normal approximation by using P(X>30.5) with
continuity correction and find the probability from the table of
standard normal distribution.

A coin is tossed 6 times. What is the
probability that the number of heads obtained will be between
2 and 3 inclusive? Express your answer as a
fraction or a decimal number rounded to four decimal places.
PLEASE DON'T ANSWER UNLESS YOU ARE CONFIDENT YOU KNOW
THE SOLUTION.

1.
two coins are tossed, find the probability that two heads are
obtained. note: each coin has two possible outcomes H (heads) and T
(tails).
2. which of these numbers cannot be a probability? why?
a) -0.00001
b) 0.5
c) 20%
d)0
e) 1
3. in a deck of 52 cards, what is the probability of drawing a
three of spades, and then a four of clubs, without
replacement?
4. what is the probability of the same outcome in #3,...

a biased coin tossed four times P(T)=2/3 x is number
of tails observed
construct the table of probabulity function f(x) and cumulative
distributive function F(x)
and the probability that at least on tail is observed ie
P(X>1)

1.)
Assume that a procedure yields a binomial distribution with a
trial repeated n=5n=5 times. Use technology to find the probability
distribution given the probability p=0.702p=0.702 of success on a
single trial.
(Report answers accurate to 4 decimal places.)
k
P(X = k)
0
1
2
3
4
5
2.)
Assume that a procedure yields a binomial distribution with a
trial repeated n=5n=5 times. Find the probability distribution
given the probability p=0.799p=0.799 of success on a single trial.
Report answers...

Let p denote the probability that a particular coin will show
heads when randomly tossed. It is not necessarily true that the
coin is a “fair” coin wherein p=1/2. Find the a posteriori
probability density function f(p|TN ) where TN is the observed
number of heads n observed in N tosses of a coin. The a priori
density is p~U[0.2,0.8], i.e., uniform over this interval. Make
some plots of the a posteriori density.

Consider the following probability distribution table:
X
1
2
3
4
P(X)
.1
.2
.3
.4
If Y = 2+3X, E(Y) is:
A.7 B.8 C.11 D.2
19.
If Y=2+3X, Var (Y) is:
A.9
B.14
C.4
D.6
20.
Which of the following about the binomial distribution is not a
true statement?
A. The probability of success is constant from trial to trial.
B. Each outcome is independent of the other. C. Each outcome is
classified as either success or failure. D....

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