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Bernoulli Trial

Each performance of an experiment with two possible outcomes is called a Bernoulli trial.
Setiap pelaksanaan percobaan dengan dua kemungkinan hasil disebut percobaan Bernoulli.
In general, a possible outcome of a Bernoulli trial is called a success or a failure.
Secara umum, hasil yang mungkin dari uji coba Bernoulli disebut sukses atau gagal.
If p is the probability of a success and q is the probability of a failure, it follows that p + q = 1.
Jika p adalah probabilitas sukses dan q adalah probabilitas kegagalan, maka p + q = 1.

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Theorem

The probability of exactly k successes in n independent Bernoulli trials, with probability of success p and probability of failure q=1−pq = 1 - p, is
Probabilitas tepat k keberhasilan dalam n uji coba Bernoulli independen, dengan probabilitas keberhasilan p dan probabilitas kegagalan q=1−pq = 1 - p, adalah
                                       C(n,k)pkqn−k~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C(n, k) p^k q^{n-k}

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Example 1

A coin is biased so that the probability of heads is 2/3. What is the probability that exactly four heads come up when the coin is flipped seven times, assuming that the flips are independent?
Koin bias sehingga probabilitas kepala adalah 2/3. Berapa probabilitas bahwa tepat empat kepala muncul ketika koin dibalik tujuh kali, dengan asumsi bahwa koin dilempar secara bebas?
Solution:
  • There are 27=1282^7 = 128 possible outcomes when a coin is flipped seven times.
  • The number of ways four of the seven flips can be heads is C(7,4)C(7, 4).
  • Because the seven flips are independent, the probability of each of these outcomes (four heads and three tails) is (2/3)4(1/3)3(2/3)^4 (1/3)^3.
  • Consequently, the probability that exactly four heads appear is

  • C(7,4)(23)4(13)3=35×1637=5602187≈0.256C(7, 4) (\dfrac{2}{3})^4 (\dfrac{1}{3})^3 = \dfrac{35 \times 16}{3^7} = \dfrac{560}{2187} \approx 0.256.

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Binomial Distribution

We denote by b(k;n,p)b(k; n, p) the probability of k successes in n independent Bernoulli trials with probability of success p and probability of failure q = 1 - p.
Kami menyatakan dengan b(k;n,p)b(k; n, p) probabilitas keberhasilan k dalam n percobaan Bernoulli independen dengan probabilitas keberhasilan p dan probabilitas kegagalan q = 1 - p.
Considered as a function of k, we call this function the binomial distribution
Dianggap sebagai fungsi k, kita menyebut fungsi ini sebagai distribusi binomial
b(k;n,p)=C(n,k)pkqn−kb(k; n,p) = C(n,k) p^k q^{n - k}

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Example 2

Suppose that the probability that a 0 bit is generated is 0.9, that the probability that a 1 bit is generated is 0.1, and that bits are generated independently. What is the probability that exactly eight 0 bits are generated when 10 bits are generated?
Misalkan probabilitas 0 bit dihasilkan adalah 0,9, probabilitas 1 bit dihasilkan adalah 0,1, dan bit dihasilkan secara independen. Berapa probabilitas bahwa tepat delapan 0 bit dihasilkan ketika 10 bit dihasilkan?
Solution:
  • By Theorem 2, the probability that exactly eight 0 bits are generated is

  • b(8;10,0.9)=C(10,8)(0.9)8(0.1)2=0.1937102445b(8; 10, 0.9) = C(10, 8) (0.9)^8 (0.1)^2 = 0.1937102445.

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Random Variables

A random variable is a function from the sample space of an experiment to the set of real numbers.
Variabel acak adalah fungsi dari ruang sampel percobaan ke himpunan bilangan real.
A random variable assigns a real number to each possible outcome.
Variabel acak memberikan bilangan real untuk setiap kemungkinan hasil.
The distribution of a random variable X on a sample space S is the set of pairs (r,p(X=r))(r, p(X = r)) for all r∈X(S)r \in X(S), where p(X=r)p(X = r) is the probability that X takes the value r.
Distribusi variabel acak X pada ruang sampel S adalah himpunan pasangan (r,p(X=r))(r, p (X = r)) untuk semua r∈X(S)r \in X (S), di mana p(X=r)p(X = r) adalah probabilitas bahwa X mengambil nilai r.

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Example 3

  • Suppose that a coin is flipped three times.
  • Let X(t)X(t) be the random variable that equals the number of heads that appear when t is the outcome.
  • Then X(t)X(t) takes on the following values:
X(HHH)=3,X(HHH) = 3,
X(HHT)=X(HTH)=X(THH)=2,X(HHT) = X(HTH) = X(THH) = 2,
X(TTH) =X(THT)=X(HTT)=1,X(TTH) ~= X(THT) = X(HTT) = 1,
X(TTT)  =0.X(TTT) ~~= 0.

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Example 4

Let XX be the sum of the numbers that appear when a pair of dice is rolled. What are the values of this random variable for the 36 possible outcomes (i,j)(i,j), where ii and jj are the numbers that appear on the first die and the second die, respectively, when these two dice are rolled?
Misalkan XX adalah jumlah angka yang muncul saat sepasang dadu dilempar. Berapa nilai variabel acak ini untuk 36 hasil yang mungkin (i,j)(i,j), di mana ii dan jj masing-masing adalah angka yang muncul pada dadu pertama dan dadu kedua, ketika kedua dadu ini dilempar?
Solution:
X((1,1))=2,X((1,1)) = 2,
X((1,2))=X((2,1))=3,X((1,2)) = X((2,1)) = 3,
X((1,3))=X((2,2))=X((3,1))=4,X((1,3)) = X((2,2)) = X((3,1)) = 4,
X((1,4))=X((2,3))=X((3,2))=X((4,1))=5,X((1,4)) = X((2,3)) = X((3,2)) = X((4,1)) = 5,
......

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Distribution of Random Variable

The distribution of a random variable XX on a sample space SS is the set of pairs (r,p(X=r))(r, p(X = r)) for all r∈X(S)r \in X(S), where p(X=r)p(X = r) is the probability that XX takes the value rr The set of pairs in this distribution is determined by the probabilities p(X=r)p(X = r) for r∈X(S)r \in X(S)

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Example 5

Suppose that a coin is flipped three times. Let X(t)X(t) be the random variable that equals the number of heads that appear when t is the outcome. Determine the distribution of X(t)X(t).
Misalkan sebuah koin dilempar tiga kali. Misalkan X(t)X(t) adalah variabel acak yang sama dengan jumlah kepala yang muncul ketika t adalah hasilnya. Tentukan distribusi X(t)X(t).
Solution:
Each of the eight possible outcomes when a fair coin is flipped three times has probability 1/8
P(X=3)=1/8,P(X=2)=3/8,P(X=1)=3/8P(X = 3) = 1/8, P(X = 2) = 3/8, P(X = 1) = 3/8, and P(X=0)=1/8P(X = 0) = 1/8
Consequently, the distribution of X(t)X(t) is the set of pairs (3,1/8)(3,1/8), (2,3/8)(2,3/8), (1,3/8)(1,3/8), and (0,1/8)(0,1/8).