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 4 times and lost 6 times.

1. Car
2. Goat
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 331 times and lost 669 times.

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