Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 5 times and lost 5 times.

1. Car
2. Car
3. Car
4. Goat
5. Goat
6. Goat
7. Goat
8. Car
9. Car
10. Goat

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 307 times and lost 693 times.

1. Car
2. Car
3. Car
4. Goat
5. Goat
6. Car
7. Goat
8. Goat
9. Car
10. Goat
11. Goat
12. Goat
13. Car
14. Goat
15. Goat
16. Goat
17. Car
18. Goat
19. Goat
20. Car
21. Goat
22. Goat
23. Goat
24. Goat
25. Goat
26. Goat
27. Goat
28. Car
29. Goat
30. Car
31. Goat
32. Car
33. Car
34. Goat
35. Goat
36. Car
37. Goat
38. Goat
39. Goat
40. Car
41. Car
42. Car
43. Car
44. Goat
45. Goat
46. Goat
47. Goat
48. Goat
49. Car
50. Goat
51. Car
52. Goat
53. Goat
54. Car
55. Goat
56. Goat
57. Goat
58. Goat
59. Car
60. Goat
61. Goat
62. Goat
63. Goat
64. Car
65. Goat
66. Car
67. Goat
68. Goat
69. Goat
70. Goat
71. Car
72. Goat
73. Goat
74. Goat
75. Goat
76. Goat
77. Car
78. Car
79. Goat
80. Goat
81. Goat
82. Goat
83. Goat
84. Goat
85. Goat
86. Car
87. Car
88. Car
89. Goat
90. Goat
91. Goat
92. Car
93. Goat
94. Goat
95. Car
96. Goat
97. Goat
98. Goat
99. Goat
100. Goat
101. Goat
102. Goat
103. Goat
104. Goat
105. Goat
106. Goat
107. Car
108. Goat
109. Car
110. Goat
111. Car
112. Car
113. Goat
114. Goat
115. Goat
116. Car
117. Car
118. Goat
119. Car
120. Goat
121. Car
122. Car
123. Goat
124. Goat
125. Goat
126. Car
127. Goat
128. Goat
129. Car
130. Car
131. Goat
132. Goat
133. Car
134. Goat
135. Car
136. Goat
137. Goat
138. Goat
139. Goat
140. Goat
141. Car
142. Car
143. Goat
144. Goat
145. Car
146. Car
147. Goat
148. Car
149. Goat
150. Goat
151. Car
152. Goat
153. Goat
154. Goat
155. Goat
156. Goat
157. Goat
158. Goat
159. Goat
160. Goat
161. Goat
162. Goat
163. Car
164. Car
165. Car
166. Goat
167. Goat
168. Goat
169. Goat
170. Goat
171. Car
172. Car
173. Goat
174. Goat
175. Goat
176. Goat
177. Goat
178. Car
179. Goat
180. Goat
181. Car
182. Goat
183. Goat
184. Car
185. Car
186. Goat
187. Car
188. Goat
189. Goat
190. Goat
191. Car
192. Goat
193. Goat
194. Car
195. Goat
196. Goat
197. Car
198. Goat
199. Car
200. Goat
201. Car
202. Goat
203. Car
204. Goat
205. Goat
206. Car
207. Goat
208. Car
209. Goat
210. Goat
211. Goat
212. Car
213. Car
214. Goat
215. Goat
216. Goat
217. Goat
218. Goat
219. Goat
220. Goat
221. Goat
222. Goat
223. Car
224. Car
225. Car
226. Goat
227. Car
228. Goat
229. Goat
230. Goat
231. Goat
232. Goat
233. Goat
234. Goat
235. Car
236. Car
237. Goat
238. Goat
239. Goat
240. Car
241. Goat
242. Goat
243. Car
244. Car
245. Goat
246. Goat
247. Car
248. Goat
249. Goat
250. Goat
251. Goat
252. Goat
253. Goat
254. Car
255. Goat
256. Car
257. Car
258. Goat
259. Goat
260. Goat
261. Car
262. Goat
263. Car
264. Car
265. Car
266. Goat
267. Car
268. Goat
269. Goat
270. Goat
271. Goat
272. Car
273. Goat
274. Car
275. Goat
276. Goat
277. Goat
278. Car
279. Goat
280. Goat
281. Car
282. Goat
283. Car
284. Car
285. Car
286. Goat
287. Goat
288. Car
289. Car
290. Goat
291. Goat
292. Goat
293. Goat
294. Car
295. Goat
296. Goat
297. Goat
298. Car
299. Goat
300. Car
301. Goat
302. Goat
303. Car
304. Goat
305. Goat
306. Goat
307. Goat
308. Goat
309. Goat
310. Car
311. Goat
312. Goat
313. Goat
314. Car
315. Goat
316. Car
317. Goat
318. Goat
319. Car
320. Car
321. Goat
322. Car
323. Goat
324. Car
325. Goat
326. Goat
327. Goat
328. Goat
329. Goat
330. Goat
331. Goat
332. Goat
333. Car
334. Goat
335. Car
336. Goat
337. Goat
338. Goat
339. Car
340. Goat
341. Goat
342. Goat
343. Car
344. Goat
345. Car
346. Goat
347. Goat
348. Goat
349. Goat
350. Car
351. Goat
352. Goat
353. Goat
354. Car
355. Goat
356. Goat
357. Car
358. Goat
359. Goat
360. Goat
361. Car
362. Car
363. Goat
364. Goat
365. Goat
366. Goat
367. Goat
368. Goat
369. Car
370. Goat
371. Car
372. Goat
373. Goat
374. Goat
375. Goat
376. Goat
377. Goat
378. Goat
379. Goat
380. Car
381. Car
382. Goat
383. Goat
384. Goat
385. Goat
386. Goat
387. Car
388. Goat
389. Goat
390. Goat
391. Car
392. Goat
393. Goat
394. Goat
395. Goat
396. Car
397. Goat
398. Goat
399. Car
400. Goat
401. Goat
402. Car
403. Car
404. Goat
405. Goat
406. Goat
407. Goat
408. Goat
409. Goat
410. Car
411. Goat
412. Car
413. Goat
414. Car
415. Goat
416. Goat
417. Goat
418. Car
419. Goat
420. Goat
421. Goat
422. Goat
423. Goat
424. Goat
425. Car
426. Car
427. Goat
428. Goat
429. Car
430. Goat
431. Goat
432. Goat
433. Goat
434. Goat
435. Goat
436. Goat
437. Goat
438. Goat
439. Goat
440. Goat
441. Goat
442. Goat
443. Goat
444. Goat
445. Goat
446. Goat
447. Car
448. Goat
449. Goat
450. Car
451. Goat
452. Goat
453. Goat
454. Goat
455. Car
456. Goat
457. Goat
458. Car
459. Car
460. Goat
461. Car
462. Goat
463. Goat
464. Car
465. Goat
466. Goat
467. Goat
468. Goat
469. Car
470. Goat
471. Car
472. Car
473. Goat
474. Goat
475. Goat
476. Goat
477. Car
478. Goat
479. Goat
480. Goat
481. Goat
482. Car
483. Goat
484. Goat
485. Goat
486. Car
487. Goat
488. Goat
489. Car
490. Car
491. Goat
492. Goat
493. Goat
494. Goat
495. Goat
496. Car
497. Car
498. Goat
499. Goat
500. Goat
501. Goat
502. Car
503. Car
504. Goat
505. Goat
506. Goat
507. Goat
508. Goat
509. Car
510. Goat
511. Goat
512. Goat
513. Goat
514. Car
515. Goat
516. Car
517. Car
518. Goat
519. Goat
520. Goat
521. Car
522. Car
523. Car
524. Goat
525. Goat
526. Car
527. Goat
528. Goat
529. Goat
530. Car
531. Car
532. Goat
533. Goat
534. Goat
535. Car
536. Goat
537. Goat
538. Car
539. Goat
540. Car
541. Car
542. Goat
543. Goat
544. Goat
545. Goat
546. Goat
547. Goat
548. Goat
549. Goat
550. Goat
551. Goat
552. Goat
553. Car
554. Car
555. Car
556. Car
557. Car
558. Goat
559. Car
560. Goat
561. Car
562. Goat
563. Car
564. Car
565. Goat
566. Goat
567. Car
568. Goat
569. Goat
570. Goat
571. Goat
572. Goat
573. Goat
574. Car
575. Goat
576. Goat
577. Goat
578. Goat
579. Goat
580. Goat
581. Goat
582. Car
583. Goat
584. Goat
585. Goat
586. Goat
587. Car
588. Goat
589. Goat
590. Goat
591. Goat
592. Goat
593. Car
594. Car
595. Goat
596. Goat
597. Car
598. Goat
599. Goat
600. Car
601. Goat
602. Goat
603. Car
604. Goat
605. Car
606. Car
607. Goat
608. Goat
609. Goat
610. Goat
611. Car
612. Car
613. Car
614. Goat
615. Goat
616. Goat
617. Goat
618. Goat
619. Goat
620. Goat
621. Car
622. Goat
623. Goat
624. Car
625. Goat
626. Goat
627. Car
628. Goat
629. Goat
630. Goat
631. Car
632. Goat
633. Goat
634. Goat
635. Goat
636. Car
637. Goat
638. Goat
639. Car
640. Goat
641. Goat
642. Goat
643. Goat
644. Goat
645. Goat
646. Goat
647. Car
648. Car
649. Goat
650. Goat
651. Goat
652. Goat
653. Car
654. Goat
655. Goat
656. Car
657. Car
658. Goat
659. Goat
660. Car
661. Goat
662. Goat
663. Goat
664. Goat
665. Goat
666. Goat
667. Goat
668. Goat
669. Car
670. Car
671. Goat
672. Car
673. Car
674. Goat
675. Goat
676. Goat
677. Goat
678. Car
679. Goat
680. Car
681. Goat
682. Goat
683. Goat
684. Goat
685. Goat
686. Goat
687. Car
688. Goat
689. Goat
690. Car
691. Goat
692. Goat
693. Car
694. Goat
695. Goat
696. Goat
697. Goat
698. Car
699. Goat
700. Goat
701. Goat
702. Goat
703. Car
704. Car
705. Goat
706. Car
707. Goat
708. Car
709. Goat
710. Goat
711. Car
712. Goat
713. Goat
714. Goat
715. Goat
716. Goat
717. Goat
718. Car
719. Goat
720. Car
721. Goat
722. Goat
723. Goat
724. Goat
725. Goat
726. Goat
727. Goat
728. Car
729. Goat
730. Car
731. Car
732. Goat
733. Car
734. Car
735. Goat
736. Car
737. Goat
738. Goat
739. Goat
740. Goat
741. Car
742. Goat
743. Goat
744. Goat
745. Car
746. Goat
747. Car
748. Goat
749. Goat
750. Car
751. Goat
752. Goat
753. Car
754. Goat
755. Goat
756. Car
757. Goat
758. Goat
759. Goat
760. Goat
761. Goat
762. Goat
763. Goat
764. Goat
765. Car
766. Goat
767. Goat
768. Car
769. Goat
770. Goat
771. Goat
772. Goat
773. Goat
774. Goat
775. Goat
776. Car
777. Goat
778. Goat
779. Goat
780. Goat
781. Goat
782. Goat
783. Car
784. Goat
785. Goat
786. Goat
787. Goat
788. Car
789. Car
790. Goat
791. Goat
792. Car
793. Goat
794. Goat
795. Car
796. Goat
797. Goat
798. Car
799. Car
800. Car
801. Goat
802. Goat
803. Goat
804. Goat
805. Goat
806. Car
807. Goat
808. Goat
809. Car
810. Car
811. Goat
812. Goat
813. Goat
814. Goat
815. Goat
816. Goat
817. Goat
818. Goat
819. Goat
820. Car
821. Goat
822. Goat
823. Car
824. Goat
825. Goat
826. Goat
827. Car
828. Car
829. Car
830. Car
831. Goat
832. Car
833. Goat
834. Goat
835. Car
836. Goat
837. Car
838. Car
839. Goat
840. Goat
841. Goat
842. Goat
843. Car
844. Goat
845. Goat
846. Car
847. Goat
848. Goat
849. Goat
850. Car
851. Goat
852. Goat
853. Goat
854. Goat
855. Goat
856. Goat
857. Goat
858. Goat
859. Goat
860. Goat
861. Car
862. Goat
863. Car
864. Car
865. Goat
866. Car
867. Goat
868. Goat
869. Car
870. Car
871. Goat
872. Goat
873. Goat
874. Car
875. Goat
876. Goat
877. Goat
878. Goat
879. Goat
880. Goat
881. Goat
882. Car
883. Goat
884. Goat
885. Car
886. Goat
887. Car
888. Goat
889. Car
890. Goat
891. Goat
892. Goat
893. Car
894. Goat
895. Goat
896. Goat
897. Car
898. Car
899. Goat
900. Goat
901. Goat
902. Goat
903. Goat
904. Car
905. Car
906. Goat
907. Car
908. Goat
909. Goat
910. Goat
911. Goat
912. Car
913. Goat
914. Goat
915. Car
916. Car
917. Goat
918. Goat
919. Goat
920. Goat
921. Car
922. Goat
923. Car
924. Goat
925. Goat
926. Goat
927. Goat
928. Goat
929. Car
930. Goat
931. Goat
932. Goat
933. Car
934. Goat
935. Car
936. Car
937. Goat
938. Car
939. Goat
940. Goat
941. Goat
942. Goat
943. Car
944. Goat
945. Goat
946. Goat
947. Car
948. Goat
949. Goat
950. Car
951. Goat
952. Goat
953. Goat
954. Car
955. Car
956. Goat
957. Goat
958. Car
959. Goat
960. Car
961. Goat
962. Car
963. Goat
964. Goat
965. Goat
966. Goat
967. Car
968. Car
969. Goat
970. Goat
971. Car
972. Car
973. Goat
974. Goat
975. Car
976. Car
977. Goat
978. Goat
979. Goat
980. Car
981. Car
982. Goat
983. Car
984. Goat
985. Car
986. Goat
987. Goat
988. Goat
989. Goat
990. Goat
991. Car
992. Goat
993. Goat
994. Goat
995. Goat
996. Car
997. Goat
998. Goat
999. Goat
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 123 times and lost 877 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 35 times and lost 65 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}